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" }} {PARA 0 "" 0 "" {TEXT -1 98 "For example, the sine and cosine function s are periodic since their values repeat at intervals of " }{XPPEDIT 18 0 "2*Pi;" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 10 "Formally, " }{XPPEDIT 18 0 "sin(t+2*Pi) = sin*t;" "6#/- %$sinG6#,&%\"tG\"\"\"*&\"\"#F)%#PiGF)F)*&F%F)F(F)" }{TEXT -1 5 " and \+ " }{XPPEDIT 18 0 "cos(t+2*Pi) = cos*t;" "6#/-%$cosG6#,&%\"tG\"\"\"*&\" \"#F)%#PiGF)F)*&F%F)F(F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 28 "It is also correct to write " }{XPPEDIT 18 0 "sin(t+4*Pi) = sin*t; " "6#/-%$sinG6#,&%\"tG\"\"\"*&\"\"%F)%#PiGF)F)*&F%F)F(F)" }{TEXT -1 65 ", so the values of the sine function also repeat at intervals of \+ " }{XPPEDIT 18 0 "4*Pi;" "6#*&\"\"%\"\"\"%#PiGF%" }{TEXT -1 12 ", but \+ since " }{XPPEDIT 18 0 "2*Pi;" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 100 " is the smallest interval over which the values repeat we call this t he period of the sine function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 11 "A function " }{XPPEDIT 18 0 "f(t)" "6#-% \"fG6#%\"tG" }{TEXT -1 11 " is called " }{TEXT 260 8 "periodic" } {TEXT -1 46 " if there is some positive number a such that " } {XPPEDIT 18 0 "f(t + a) = f(t)" "6#/-%\"fG6#,&%\"tG\"\"\"%\"aGF)-F%6#F (" }{TEXT -1 17 " for all numbers " }{TEXT 271 1 "t" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 20 "The smallest number " }{TEXT 272 1 "p" } {TEXT -1 11 " for which " }{XPPEDIT 18 0 "f(t + p) = f(t)" "6#/-%\"fG6 #,&%\"tG\"\"\"%\"pGF)-F%6#F(" }{TEXT -1 9 " for all " }{TEXT 273 1 "t " }{TEXT -1 16 ", is called the " }{TEXT 260 6 "period" }{TEXT -1 4 " \+ of " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "For another exa mple, the function " }{XPPEDIT 18 0 "f(t)=cos^2*t" "6#/-%\"fG6#%\"tG*& %$cosG\"\"#F'\"\"\"" }{TEXT -1 25 " is periodic with period " } {XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 23 ". 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<*F,$\"3e(3yy)3JJuF,7$$\"3emm;zihl&*F,$\"33&4*ecU'3z(F,7$$\"39LLL3#G,* **F,$\"35HW!o*f[l\")F,7$$\"3\\k***H&)F,7$$\"3!*** *\\PQ#\\\"3\"Fcs$\"3h/DD\\I1>))F,7$$\"3BLL$e\"*[H7\"Fcs$\"3Qj8;@#eW4*F ,7$$\"3#*******pvxl6Fcs$\"3)[+'*GfMAL*F,7$$\"3z****\\_qn27Fcs$\"3qD!zH $)RP^*F,7$$\"3%)***\\i&p@[7Fcs$\"3IVQmY+1Q'*F,7$$\"3%)**\\(=GB2F\"Fcs$ \"3'RTKt0DZo*F,7$$\"3#)****\\2'HKH\"Fcs$\"3Ig=\"HE%Q:(*F,7$$\"3=LL3UDX 88Fcs$\"3rYUKXmSH(*F,7$$\"3_mmmwanL8Fcs$\"3fMvq&ea3t*F,7$$\"39LL$exn_N \"Fcs$\"3\"[\"fMq()))=(*F,7$$\"3'******\\2goP\"Fcs$\"3#3d(3?\"yMp*F,7$ $\"3CLLeR<*fT\"Fcs$\"3Za7sk%4`h*F,7$$\"3'******\\)Hxe9Fcs$\"3#y7!3)o3u [*F,7$$\"3Ymm\"H!o-*\\\"Fcs$\"3]XZB%*)F,7$$\"3\"****\\i!*3 `i\"Fcs$\"3(oR;pH,/s)F,7$$\"3QLLL$*zym;Fcs$\"3%3T&zR@.1&)F,7$$\"3GLL$3 N1#4Fcs$\"3'\\euIt)*)) f(F,7$$\"3/++v.Uac>Fcs$\"3_P`Yf;+\\vF,7$$\"\"#F)$\"3U>Lqe/,EvF,-%'COLO URG6&%$RGBG$\"*++++\"!\")F(F(-F$6$7V7$$!\"#F)F*7$$!3ymmm\"p0k&>Fcs$\"3 klI(3i,9G$F,7$$!3FLL3FcsF67$$!3cmm;Wp\"e(=Fcs$\"3'Gpx)e^R!o$F,7$$ !3hmm;4m(G$=FcsF@7$$!3QLL3i.9!z\"FcsFE7$$!3emmT!R=0v\"Fcs$\"3]bj%4e#4Y SF,7$$!3)****\\P8#\\4\\`I7%F,7$$!3!om;/siqm\"Fcs$\"3m=DZN 2*R>%F,7$$!3%****\\(y$pZi\"Fcs$\"3[xu8IjxkUF,7$$!3ILLLyaE\"e\"FcsFhn7$ $!3mmm;>s%Ha\"FcsF]o7$$!3/+++N*4)*\\\"FcsFbo7$$!3-+++Db\\c9FcsFgo7$$!3 *)*****\\1aZT\"Fcs$\"3A(ysH%>$**)[F,7$$!3ommT?)[oP\"FcsFap7$$!3ZLLL=ex J8Fcs$\"3/&fXgR\"*pN&F,7$$!3SLLLtIf$H\"FcsF[q7$$!3;++vju<\\7FcsF`q7$$! 3aLLLB@')47FcsFeq7$$!3'****\\P'psm6FcsFjq7$$!35++D\"4_c7\"Fcs$\"3M^TDt \"yd.(F,7$$!3ULL$3x%z#3\"FcsFdr7$$!3MLL3s$QM/\"FcsFir7$$!3pmm;zr)4+\"F csF^s7$$!3Iom;/K#*o&*F,Fds7$$!3-,+]ih2&=*F,Fis7$$!3snmmT3^q()F,F^t7$$! 3q++++VAU$)F,Fct7$$!33-++v%HK#zF,Fht7$$!3d,+]P/$y^(F,F]u7$$!3o,+D\"=nF H(F,Fbu7$$!3y,++DRqnqF,Fgu7$$!3Eom;zXZloF,F\\v7$$!3uMLLL_CjmF,Fav7$$!3 cnmmTAKZkF,Ffv7$$!3R+++]#*RJiF,F[w7$$!3enm;/E3SeF,F`w7$$!3M+++],F7aF,F ew7$$!3VNL$3(>t4]F,Fjw7$$!3?,+](ej*)e%F,F_x7$$!3=MLL$e&exTF,Fdx7$$!3#4 ++v$4\"pu$F,Fix7$$!3;mmmm+7KLF,F^y7$$!3Bnmm\"\\Oz!HF,Fcy7$$!3mLL$3Pls[ 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7$F(F`\\oFc_oFh[lFg_o-F$6&F`aoF]`oFh[lFg_o-F$6&F`aoFb`oFh[lFg_o-F$6&7# 7$Fd[lFi`oFh[lFg_oFc_o-%)POLYGONSG6&7&7$$\"+++++r!#5$!\"%!\"\"7$$\"+++ ++pFaboFbbo7$Ffbo$!++++]AFabo7$F_boFibo7%7$$\"++++]iFaboFibo7$$\"\"(Fd bo$!#:Fd\\l7$$\"++++]xFaboFibo-Fh_o6#%,PATCHNOGRIDG-%&COLORG6&F[\\l$Fe _nFdboF)$\"\"*Fdbo-F[bo6&7&7$$\"++++5Z!\"*Fbbo7$$\"++++!p%FgdoFbbo7$Fi doFibo7$FedoFibo7%7$$\"++++DYFgdoFibo7$$\"#ZFdboFcco7$$\"++++vZFgdoFib oFhcoF[do-F$6$7$7$F\\^o$!3A+++++++SF,7$Fj^oF[foF[do-%%TEXTG6%7$F($!\"& Fd\\lQ\"a6\"-F\\do6&F[\\lF)F)$\"\"\"Fd\\l-F_fo6%7$Fd[lFbfoQ\"bFefoFffo -F_fo6%7$Faco$!\"(Fd\\lQ'u=t-2pFefoFa^o-F_fo6%7$FbeoFagoQ\"tFefoFa^o-% *AXESSTYLEG6#%%NONEG-%+AXESLABELSG6%Q\"xFefoQ!Fefo-%%FONTG6#%(DEFAULTG -%%VIEWG6$;Fc\\lFc[oFdho" 1 2 0 1 10 0 2 9 1 1 2 1.000000 46.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Cu rve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve \+ 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" "Curve 25" "Curve 26" "Curve 27" "Curve 28" "Curve 29" }}{TEXT -1 3 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "For an ex ample, consider the function " }{XPPEDIT 18 0 "g(t)" "6#-%\"gG6#%\"tG " }{TEXT -1 25 " defined on the interval " }{XPPEDIT 18 0 "0<=t" "6#1 \"\"!%\"tG" }{XPPEDIT 18 0 "`` < 3;" "6#2%!G\"\"$" }{TEXT -1 6 " by: \+ " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g(t) = PIECEWISE([t, 0 <= t and t < 1],[1, 1 <= t and \+ t < 2],[3-t, 2 <= t and t < 3]);" "6#/-%\"gG6#%\"tG-%*PIECEWISEG6%7$F' 31\"\"!F'2F'\"\"\"7$F031F0F'2F'\"\"#7$,&\"\"$F0F'!\"\"31F5F'2F'F8" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "g(t)" "6#-%\"gG6#%\"tG" } {TEXT -1 40 " can be extended to a periodic function " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 16 " with period 3. " }}{PARA 0 "" 0 "" {TEXT -1 10 "We write: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "f(t) = PIECEWISE([t, 0 <= t and t < 1],[1, 1 <= t and t < 2],[3-t, 2 <= t and t < 3]);" "6#/-%\"fG6#%\"tG-%*PIECEWISEG6%7$F'3 1\"\"!F'2F'\"\"\"7$F031F0F'2F'\"\"#7$,&\"\"$F0F'!\"\"31F5F'2F'F8" } {TEXT -1 3 " , " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 " and " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 28 " is periodic with period 3. " }}{PARA 256 "" 0 "" {GLPLOT2D 619 130 130 {PLOTDATA 2 "6)-%'CURVESG6%7W7$$\"\"!F)F(7$$\"3s******\\i9Rl!# >F+7$$\"3/++vVA)GA\"!#=F/7$$\"3+++]Peui=F1F37$$\"3A++]i3&o]#F1F67$$\"3 %)***\\(oX*y9$F1F97$$\"3z***\\P9CAu$F1F<7$$\"3!)***\\P*zhdVF1F?7$$\"31 ++v$>fS*\\F1FB7$$\"3$)***\\(=$f%GcF1FE7$$\"3Q+++Dy,\"G'F1FH7$$\"33++]7 &) \\\"FinFjn7$$\"3)***\\P>:mk:FinFjn7$$\"3'***\\iv&QAi\"FinFjn7$$\"31++v tLU%o\"FinFjn7$$\"3!******\\Nm'[FinFjn7$$\"3)***\\78)y,(>FinFjn7 $$\"3z*****\\@80+#Fin$\"35-++]y'[***F17$$\"3q***\\Pm,H.#Fin$\"3-.+]iL) 4n*F17$$\"31++]7,Hl?Fin$\"3[*****\\())4Z$*F17$$\"3()**\\P4w)R7#Fin$\"3 Q,+D1R7g()F17$$\"3;++]x%f\")=#Fin$\"3G)****\\A0%=\")F17$$\"3!)**\\P/-a [AFin$\"3/-+Dczf9vF17$$\"3/+](=Yb;J#Fin$\"3g***\\7QXM)oF17$$\"3')**** \\i@OtBFin$\"3G,++v$yjE'F17$$\"3')**\\PfL'zV#Fin$\"3R,+D1kO?cF17$$\"3> +++!*>=+DFin$\"37)******4!=)*\\F17$$\"3-++DE&4Qc#Fin$\"3u****\\PZ!>O%F 17$$\"3=+]P%>5pi#Fin$\"3E)**\\i0)*3t$F17$$\"39+++bJ*[o#Fin$\"3e)***** \\%o5:$F17$$\"33++Dr\"[8v#Fin$\"39****\\(G=l[#F17$$\"3++++Ijy5GFin$\"3 +++++n8#*=F17$$\"31+]P/)fT(GFin$\"3G***\\i&>Se7F17$$\"31+]i0j\"[$HFin$ \"3V$***\\P%p$=lF-7$$\"\"$F)F(-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-%*T HICKNESSG6#F[o-F$6%7gpFdw7$$\"31++]i9RlIFin$\"3F0++]i9RlF-7$$\"36+++DH yIJFin$\"30,++]#HyI\"F17$$\"39+](oozw=$Fin$\"3S,+voozw=F17$$\"3=++v[kd WKFin$\"3w,+]([kdW#F17$$\"3.+]73Gc3LFin$\"3K++D\"3Gc3$F17$$\"3))****\\ n\"\\DP$Fin$\"3*))****\\n\"\\DPF17$$\"3C+++q'fpV$Fin$\"3W-+++nfpVF17$$ \"3;++]s,P,NFin$\"3b,++D++vj=pDTFinFjn7$$\"3')*****\\c.iD%FinFjn7$$\"3C++ ]U$e6P%FinFjn7$$\"3))*****\\>q0]%FinFjn7$$\"3=+++DM^IYFinFjn7$$\"3))** ***\\!ytbZFinFjn7$$\"3?++vQNXp[FinFjn7$$\"3Q+DJ!HeK!\\FinFjn7$$\"3n** \\(=/jq$\\FinFjn7$$\"3K\\il1&Fin$\"3G&****\\PL0Q *F17$$\"3.+++!y?#>^Fin$\"3u*******>#z2))F17$$\"3)***\\P%>We=&Fin$\"31+ +Dc!e:9)F17$$\"3'****\\(3wY__Fin$\"3S++]7RKvuF17$$\"3M+]P>1W6`Fin$\"3i '**\\i!Qf&)oF17$$\"3#)******HOTq`Fin$\"3s,+++P'eH'F17$$\"3v**\\Ppj6NaF in$\"3W-+D1j$)[cF17$$\"3o***\\(3\">)*\\&Fin$\"3:.+]7*3=+&F17$$\"3q**** \\<9VhbFin$\"3/.++Deo&Q%F17$$\"3r***\\isVIi&Fin$\"3#H++vti&pPF17$$\"3] **\\(oqHto&Fin$\"3)\\+]7$HqEJF17$$\"3=++](o:;v&Fin$\"3;)****\\7VQ[#F17 $$\"3(***\\i&G]1\"eFin$\"3K++vVr\\$*=F17$$\"3k++v$)[opeFin$\"3j$***\\i 6:.8F17$$\"3t+]7t;OLfFin$\"3aE**\\(oKQm'F-7$$\"3%*****\\i%Qq*fFin$\"3W b+++v`hH!#?7$$\"3]**\\i]2=jgFin$\"3V]**\\i]2=jF-7$$\"3&****\\(QIKHhFin $\"3]****\\(QIKH\"F17$$\"3Q+++&4+p='Fin$\"3w.++]4+p=F17$$\"3#****\\7:x WC'Fin$\"38****\\7:xWCF17$$\"3-+]P\\>m1jFin$\"3?++v$\\>m1$F17$$\"37++] Zn%)ojFin$\"3E,++vuY)o$F17$$\"3t***\\(G(*3LkFin$\"3O(***\\(G(*3L%F17$$ \"3C+++5FL(\\'Fin$\"3M-+++rKt\\F17$$\"3-++vL>=glFin$\"3F++]P$>=g&F17$$ \"3#)****\\d6.BmFin$\"3A)****\\d6.B'F17$$\"3!***\\785%Qo'Fin$\"3(*)** \\785%QoF17$$\"3(****\\(o3lWnFin$\"3s****\\(o3lW(F17$$\"3<+]iX)p@\"oFi n$\"3k,+Dc%)p@\")F17$$\"3O++]A))ozoFin$\"3a.++D#))oz)F17$$\"3(****\\ii d.%pFin$\"3n****\\iid.%*F17$$\"3e******Hk-,qFinFjn7$$\"3S****\\FL!e1(F inFjn7$$\"36+++D-eIrFinFjn7$$\"3u***\\(=_(zC(FinFjn7$$\"3M+++b*=jP(Fin Fjn7$$\"3g***\\(3/3(\\(FinFjn7$$\"33++vB4JBwFinFjn7$$\"3u*****\\KCnu(F inFjn7$$\"3s***\\(=n#f(yFinFjn7$$\"3g**\\P\\`9QzFinFjn7$$\"3\\******zR O+!)Fin$\"390+++-O'***F17$$\"3)))**\\i^\"*R1)Fin$\"3=6+]P[3g$*F17$$\"3 0++]_!>w7)Fin$\"3[*****\\Z4Qs)F17$$\"33,]i?(>2>)Fin$\"37*)*\\Pz-G4)F17 $$\"3O++v)Q?QD)Fin$\"3_'***\\7hzhuF17$$\"3@,]P\\L!=J)Fin$\"3'z)*\\i]m> )oF17$$\"3G+++5jyp$)Fin$\"3;(*******o8-jF17$$\"3B++DE8CO%)Fin$\"3s(*** \\PnePcF17$$\"3<++]Ujp-&)Fin$\"3I)****\\dOI(\\F17$$\"33++D,X8i&)Fin$\" 39****\\()\\lyVF17$$\"3++++gEd@')Fin$\"3+++++MF%y$F17$$\"31+]PMh%\\o)F in$\"3G***\\ilQ0:$F17$$\"39++v3'>$[()Fin$\"3d)***\\7R!o^#F17$$\"3p**** **4h(*3))Fin$\"32.+++*Q-\">F17$$\"3-,+D6Ejp))Fin$\"3!)*)**\\()Qn.8F17$ $\"3^+]i0j\"[$*)Fin$\"3-\\**\\P%p$=lF-7$$\"\"*F)F(-Fhw6&Fjw$\")#)eqkF] x$\"))eqk\"F]xFgamF^x-%%TEXTG6$7$$\"#$*!\"\"$F_bmF_bmQ\"t6\"-Fjam6$7$$ !\"$F_bm$\"#8F_bmQ%f(t)Fbbm-%*AXESTICKSG6$%(DEFAULTG\"\"#-%+AXESLABELS G6%%!GFccm-%%FONTG6#F^cm-%%VIEWG6$;FfbmF]bm;F`bmFhbm" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3 " "Curve 4" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "The Maple function: " }{TEXT 0 5 "floor" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "The Maple function " } {TEXT 0 5 "floor" }{TEXT -1 28 " applied to a real constant " }{TEXT 296 1 "x" }{TEXT -1 30 " gives has the value which is " }{TEXT 260 44 "the greatest integer less than or equal to x" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 21 "For each real number " }{TEXT 293 1 "t" } {TEXT -1 15 ", the floor of " }{TEXT 294 1 "t" }{TEXT -1 37 " is the n earest integer neighbour to " }{TEXT 295 1 "t" }{TEXT -1 36 " on the l eft (less than or equal to " }{TEXT 292 1 "t" }{TEXT -1 2 ")." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "For examp le, since " }{XPPEDIT 18 0 "sqrt(8);" "6#-%%sqrtG6#\"\")" }{TEXT -1 1 " " }{TEXT 259 1 "~" }{TEXT -1 15 " 2.828427125, " }{XPPEDIT 18 0 "fl oor(sqrt(8)) = 2;" "6#/-%&floorG6#-%%sqrtG6#\"\")\"\"#" }{TEXT -1 7 ", but " }{XPPEDIT 18 0 "floor(-sqrt(8)) = -3;" "6#/-%&floorG6#,$-%%sqr tG6#\"\")!\"\",$\"\"$F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "floor(sqrt(8));\nfloor( -sqrt(8));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 31 "At least for a positive number " }{TEXT 290 1 "t" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "floor(t)" "6#-%&floorG6#%\"tG" } {TEXT -1 25 " is sometimes called the " }{TEXT 260 12 "integer part" } {TEXT -1 4 " of " }{TEXT 291 1 "t" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 92 "Although the floor function is not periodic, it can be us ed to construct periodic functions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "Example 1: square wave" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 18 "For example, let \+ " }{XPPEDIT 18 0 "f(t) = (-1)^floor(t);" "6#/-%\"fG6#%\"tG),$\"\"\"!\" \"-%&floorG6#F'" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 6 "Since \+ " }{XPPEDIT 18 0 "floor(t)" "6#-%&floorG6#%\"tG" }{TEXT -1 50 " is alw ays an integer the only possible values of " }{XPPEDIT 18 0 "f(t)" "6# -%\"fG6#%\"tG" }{TEXT -1 14 " are 1 and -1." }}{PARA 0 "" 0 "" {TEXT -1 13 "The value of " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 15 " is 1 whenever " }{XPPEDIT 18 0 "floor(t)" "6#-%&floorG6#%\"tG " }{TEXT -1 22 " is even, and -1 when " }{XPPEDIT 18 0 "floor(t)" "6#- %&floorG6#%\"tG" }{TEXT -1 8 " is odd." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(t) = PIECEWISE( [1, `when `*floor(t)*` is even`],[-1, `when `*floor(t)*` is odd`]);" " 6#/-%\"fG6#%\"tG-%*PIECEWISEG6$7$\"\"\"*(%&when~GF,-%&floorG6#F'F,%)~i s~evenGF,7$,$F,!\"\"*(F.F,-F06#F'F,%(~is~oddGF," }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "f(t) = 1;" "6#/-%\"fG6#%\"tG\"\"\"" }{TEXT -1 6 " when \+ " }{TEXT 289 1 "t" }{TEXT -1 112 " sits between a pair of consecutive \+ integers in which the first integer is even and the second integer is \+ odd. \n" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(t) = -1;" "6#/-%\"fG6#%\"tG,$\"\"\"!\"\"" }{TEXT -1 118 " when t sits between a \+ pair of consecutive integers in which the first integer is odd and the second integer is even. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "f := t -> (-1)^floor(t);\nplot(f(t ),t=0..8.5,-1.2..1.2,thickness=2,\n color=COLOR(RGB,0,.5,.2),la bels=[t,`f(t)`],ytickmarks=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" fGf*6#%\"tG6\"6$%)operatorG%&arrowGF()!\"\"-%&floorG6#9$F(F(F(" }} {PARA 13 "" 1 "" {GLPLOT2D 595 147 147 {PLOTDATA 2 "6(-%'CURVESG6#7es7 $\"\"!$\"\"\"F(7$$\"1nm;/\"eF&=!#;F)7$$\"1M$eR-L[Y$F.F)7$$\"1nm\"H()zx F&F.F)7$$\"1omT5Tu-rF.F)7$$\"1,v$4O*)3,)F.F)7$$\"1L$e9hM!>*)F.F)7$$\"1 mmTg!=+M*F.F)7$$\"1+]P4:+h(*F.F)7$$\"1%e9;PZi')*F.F)7$$\"1nT&QB$\\r**F .F)7$$\"1'R(\\;;T-5!#:$!\"\"F(7$$\"1v$4'4Rn25FMFN7$$\"1a8s-i$H,\"FMFN7 $$\"1LL$e\\)>=5FMFN7$$\"1]7GowCR5FMFN7$$\"1n\"H2%oHg5FMFN7$$\"1Le9OwZZ 6FMFN7$$\"1+DcJ%eYB\"FMFN7$$\"1n\"H#QM)\\T\"FMFN7$$\"1+DJS,t%f\"FMFN7$ $\"1LL3<FMFN7$$\"14_v+5Rl>FM FN7$$\"1](oHo2$))>FMFN7$$\"1O@_`o.%*>FMFN7$$\"1@b2Cgw**>FMFN7$$\"12*GY >&\\0?FMF)7$$\"1#H#=lVA6?FMF)7$$\"1i!*G1FoA?FMF)7$$\"1LeRZ59M?FMF)7$$ \"1_!)GFMF)7$$\"1L$3_5\"4@HFMF)7$$\"1$e*[)ov8%HFMF)7 $$\"1M3xr-mhHFMF)7$$\"1f9TjD!=(HFMF)7$$\"1%3_]&[%>)HFMF)7$$\"1'Rs3+;q) HFMF)7$$\"13FpYr3#*HFMF)7$$\"1@I^#Her*HFMF)7$$\"1LLLQ%HA+$FMFN7$$\"1$FMFN7$$\"1LL$e(f3eLFMFN7$$\"1+D1//TTNFMFN7$$ \"1+v=7'yfr$FMFN7$$\"1nd,&FMFN7$$\"1vV ts>L@]FMFN7$$\"1QMxA:YK]FMFN7$$\"1+D\"G2\"fV]FMFN7$$\"1](oHF4\")3&FMFN 7$$\"1+]7tuiK^FMFN7$$\"1]PfoZx=_FMFN7$$\"1,D1k?#\\I&FMFN7$$\"1+](=$eA' \\&FMFN7$$\"1LL$ex?\"ocFMFN7$$\"1n;HZ!)))fdFMFN7$$\"1,+v=`l^eFMFN7$$\" 1$zpBrKK*eFMFN7$$\"1%e*)f55[$fFMFN7$$\"1![*z-))fbfFMFN7$$\"1v$4'*\\(Qw fFMFN7$$\"1AV,[=y')fFMFN7$$\"1r#>k>wr*fFMFN7$$\"1XU# [aqv+'FMF)7$$\"1$pE!>xw7gFMF)7$$\"1m\"HK*['z,'FMF)7$$\"1$eR(R]()3hFMF) 7$$\"1,+D'=&y*>'FMF)7$$\"1LeR7R'3P'FMF)7$$\"1+DJv/p\\lFMF)7$$\"1LL3Fh_ CnFMF)7$$\"1;HK*[Wg\"oFMF)7$$\"1+Dc^Gc2pFMF)7$$\"1U&)e\"QM;&pFMF)7$$\" 1%e9;\"fq&*pFMF)7$$\"1Sy'G&[@,qFMFN7$$\"1%4@TzBn+(FMFN7$$\"1\\VPNFB7qF MFN7$$\"10wiw;u2wFMFN7$$\"1M$3_['[&z(FMFN7$$\"1,vVV0pz yFMFN7$$\"1nmm,Y*Q'zFMFN7$$\"12Rv()p6vzFMFN7$$\"1Y6%QPRj)zFMFN7$$\"1mZ )oc]>*zFMFN7$$\"1&QG*f=*G7.)FMF)7$$\"1$ek.ptO0)FMF)7$$\"1TNrMKc)4)FMF)7$$\"1+D1zFXV \")FMF)7$$\"1++]AGQH#)FMF)7$$\"1+v$f'GJ:$)FMF)7$$\"1+++++++&)FMF)-%*AX ESTICKSG6$%(DEFAULTG\"\"$-%*THICKNESSG6#\"\"#-%&COLORG6&%$RGBGF($\"\"& FO$FielFO-%+AXESLABELSG6$%\"tG%%f(t)G-%%VIEWG6$;F($\"#&)FO;$!#7FO$\"#7 FO" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 27 " is periodic with period 2 ." }}{PARA 0 "" 0 "" {TEXT -1 10 "This is a " }{TEXT 260 11 "square wa ve" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "An alternative way of describing " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 40 " is to give the piecwise description o f " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 63 " over the in terval from 0 to 2 and then prescribe the value of " }{XPPEDIT 18 0 "f (t)" "6#-%\"fG6#%\"tG" }{TEXT -1 26 " elsewhere by saying that " } {XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 27 " is periodic with period 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(t) = PIECEWISE([1, 0 <= t and t < 1],[-1, 1 <= t and t < 2]);" "6#/-%\"fG6#%\"tG-%*PIECEWISEG6$7$\"\"\"31\"\"!F'2 F'F,7$,$F,!\"\"31F,F'2F'\"\"#" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 " and " }{XPPEDIT 18 0 "f (t);" "6#-%\"fG6#%\"tG" }{TEXT -1 28 " is periodic with period 2. 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\"z'FjtF+7$$\"1w1PWWl7oFjtF+7$$\"1M;L()RRLoFjtF+7$$\"1xx[lJgaoFjtF+7$$ \"100%4sOc(oFjtF+7$$\"1(*4%RPk\\*oFjtF+7$$\"1$3/a/;r\"pFjtF+7$$\"1hEVk (Gp$pFjtF+7$$\"13&*HbK0epFjtF+7$$\"1\"Q$[)3s#ypFjtF+7$$\"1+++')******p FjtF+F[uFbu-%*AXESTICKSG6$%(DEFAULTG\"\"$-%+AXESLABELSG6$%\"tG%%f(t)G- %%VIEWG6$;F-$\"#&)F]v;$!#7F]v$\"#7F]v" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "In graphing such piece wise functions the vertical lines are often left in the picture." }} {PARA 0 "" 0 "" {TEXT -1 254 "On the one hand it seems to be easier on the eye to follow the graph from left to right in this form, and the \+ period can easily be seen. 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(**F17$$\"3)***\\iS![))*RFjp$\"3))***\\iS![))**F17$$\"3a6O#RG:1+%Fjp$ \"31Ea6O#RG:'Fgw7$$\"3@AAAFDQ-SFjp$\"3E3AAAs_#Q#F]q7$$\"3#)*****\\dt$G SFjp$\"3x\")*****\\dt$GF-7$$\"3UxxxAYOaSFjp$\"3sUxxxAYOaF-7$$\"3tLLL`e x4TFjp$\"3GPLLL&ex4\"F17$$\"31+++!HlZ;%Fjp$\"3k++++HlZ;F17$$\"3zWWW%*o eBUFjp$\"3&zWWW%*oeB#F17$$\"3#=AAAT%\\vUFjp$\"3==AAAT%\\v#F17$$\"3m@AA 7\\QNVFjp$\"3m;AAA\"\\QN$F17$$\"33666r9V*Q%Fjp$\"3%4666r9V*QF17$$\"3z5 66;x!HW%Fjp$\"3%z566;x!HWF17$$\"3i))))))e=T+XFjp$\"3<'))))))e=T+&F17$$ \"3Yxxx<'[\"eXFjp$\"3ouxxxh[\"e&F17$$\"3CWWW>0o5YFjp$\"3OUWW%>0o5'F17$ $\"3cbbblqYmYFjp$\"3sbbbb1nkmF17$$\"3GAAA#G44s%Fjp$\"3\"GAAA#G44sF17$$ \"3QWWW>_#*zZFjp$\"3sVWW%>_#*z(F17$$\"3zKLL8/*4$[Fjp$\"3*yKLL8/*4$)F17 $$\"31+++X1k*)[Fjp$\"3k+++]kS'*))F17$$\"3,666Y\\xW\\Fjp$\"39566h%\\xW* F17$$\"3/bb0Ge2s\\Fjp$\"3O]bb!Ge2s*F17$$\"3%*******4nP**\\Fjp$\"3Y**** ***4nP***F17$$\"3Q****\\sV3,]Fjp$\"3Ov$****\\sV3\"F]q7$$\"3p*****\\.#z -]Fjp$\"3'Hp*****\\.#z#F]q7$$\"3,++](p*\\/]Fjp$\"3e5+++vp*\\%F]q7$$\"3 W******ft?1]Fjp$\"3**R%******ft?'F]q7$$\"33+++&oA'4]Fjp$\"3@v+++]oA'*F ]q7$$\"3#)******4!QI,&Fjp$\"3A#)******4!QI\"F-7$$\"3?+++g'o)>]Fjp$\"3Y ?+++g'o)>F-7$$\"3q******4$*pE]Fjp$\"3')p******4$*pEF-7$$\"3d******41OS ]Fjp$\"3\\d******41OSF-7$$\"3Y******4>-a]Fjp$\"37X******4>-aF-7$$\"3-+ +]K#*)=3&Fjp$\"3+-++]K#*)=)F-7$$\"3q*****\\bc(4^Fjp$\"3+(*****\\bc(4\" F17$$\"3g*****\\MM)o^Fjp$\"33'*****\\MM)o\"F17$$\"3zKLL$*G+B_Fjp$\"3'y KLL$*G+B#F17$$\"3u5661UZw_Fjp$\"3K266h?ukFF17$$\"3IWWWfTjL`Fjp$\"3+VWW %fTjL$F17$$\"3%\\bb0op1R&Fjp$\"3b\\bb0op1RF17$$\"3%zxxxWi=W&Fjp$\"3Xzx xxWi=WF17$$\"3++++++++bFjp$\"3++++++++]F1-%+AXESLABELSG6$%\"tG%%f(t)G- %*THICKNESSG6#\"\"#-%*AXESTICKSG6$%(DEFAULTGF]im-%&COLORG6&%$RGBG$\"\" '!\"\"FfimF)-%%VIEWG6$;F($\"#bFhim;$!\"#Fhim$\"#7Fhim" 1 2 0 1 10 2 2 6 1 4 2 1.000000 44.000000 44.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "This function " } {XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 27 " is periodic with period 1." }}{PARA 0 "" 0 "" {TEXT -1 8 "It is a " }{TEXT 260 13 "saw tooth wave" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "f(t) =t" "6#/-%\"fG6#%\"tGF'" }{TEXT -1 5 ", " } {XPPEDIT 18 0 "0<=t" "6#1\"\"!%\"tG" }{XPPEDIT 18 0 "``<1" "6#2%!G\"\" \"" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 5 " and " }{XPPEDIT 18 0 "f(t);" "6#-%\"fG6#%\"tG" }{TEXT -1 28 " is periodic with period \+ 1. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "Example 3: triangle wave" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "An example of a " }{TEXT 260 15 "triangular wave" }{TEXT -1 43 " can be obtained by the following function:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "f(t ) = PIECEWISE([t-floor(t), `when `*floor(t)*` is even`],[1-(t-floor(t) ), ` when `*floor(t)*` is odd`]);" "6#/-%\"fG6#%\"tG-%*PIECEWISEG6$7$, &F'\"\"\"-%&floorG6#F'!\"\"*(%&when~GF--F/6#F'F-%)~is~evenGF-7$,&F-F-, &F'F--F/6#F'F1F1*(%'~when~GF--F/6#F'F-%(~is~oddGF-" }{TEXT -1 2 " ." } }{PARA 0 "" 0 "" {TEXT -1 27 "\nAn alternative formula is:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(t) = 1/2+(t-floor(t)-1/2) *(-1)^floor(t);" "6#/-%\"fG6#%\"tG,&*&\"\"\"F*\"\"#!\"\"F**&,(F'F*-%&f loorG6#F'F,*&F*F*F+F,F,F*),$F*F,-F06#F'F*F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "f := t -> 1/2+(t-floor(t)-1/2)*(-1)^floor(t);\nplot(f(t),t=0..6.5,-0.2. .1.2,thickness=2,numpoints=100,\n color=COLOR(RGB,.9,.4,0),labels= [t,`f(t)`],ytickmarks=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6 #%\"tG6\"6$%)operatorG%&arrowGF(,&#\"\"\"\"\"#F.*&,(9$F.-%&floorG6#F2! \"\"#F.F/F6F.)F6F3F.F.F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 604 179 179 {PLOTDATA 2 "6(-%'CURVESG6#7^s7$$\"\"!F)F(7$$\"3CihhhhSpo!#>$\"3-l hhhhSpoF-7$$\"3OBBBtRk%G\"!#=F17$$\"3U5555R#o&>F3F57$$\"3E#>>>\\fMj#F3 F87$$\"3>$HHH9zoI$F3F;7$$\"3@888j`AJRF3F>7$$\"3uggg5(*pxXF3FA7$$\"3ubb b0WGY_F3FD7$$\"3&4444%\\s7fF3FG7$$\"3K[ZZZ4C)f'F3FJ7$$\"30?>>>Q/-sF3FM 7$$\"3?WUUUGy\")yF3FP7$$\"3!*RRRRHJk&)F3FS7$$\"3Gfdddr0A#*F3FV7$$\"3z: 99RXq?&*F3FY7$$\"3>rqq?>N>)*F3Ffn7$$\"3VddKw?83**F3Fin7$$\"3yWW%>B7p** *F3F\\o7$$\"3@8jvQ#p&35!#<$\"3(y'oV7wI9**F37$$\"3%===VDZu,\"Fao$\"3j\" ==oXFb#)*F37$$\"3J>>W&G._.\"Fao$\"3#p!3eXr'zk*F37$$\"3yccc;$fH0\"Fao$ \"3AKMMMoSq%*F37$$\"3LEEEmg786Fao$\"3mOPPP$R(o))F37$$\"3maaa*H9J=\"Fao $\"3Q`aa/q&)o\")F37$$\"3;000Xd1X7Fao$\"3W[\\\\\\DM\\vF37$$\"3waaa*HOIJ \"Fao$\"3W_aa/qjpoF37$$\"3I111T.wx8Fao$\"3.PRR*e'RAiF37$$\"3kQQQ)3$HX9 Fao$\"3o8;;;\"pqa&F37$$\"3:(oo=B4t]\"Fao$\"3^GJJ\"o2p#\\F37$$\"3#3222- -Ud\"Fao$\"3u\"HHHzzzD%F37$$\"3\\VVVG[oV;Fao$\"33lll:<:jNF37$$\"3.#==o \")pTq\"Fao$\"3uz\"==$=IeHF37$$\"3iJJJh_\\pFao$\"37PXXXNP*p*F-7$$\"3CFF x%zY\\$>Fao$\"3!fFFF_?`]'F-7$$\"33+++Dt)o'>Fao$\"3[#******\\n7J$F-7$$ \"3aIIb\"e>Y)>Fao$\"3$o%ppW=/Q:F-7$$\"3*411\"Q=N-?Fao$\"3K))411\"Q=N#! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 105 "This triangular wave can also be obtain ed by writing a Maple procedure, based on the first formula above." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "g := proc(t)\n local flr,frac;\n flr := floor(t);\n frac := t-flr; \n if irem(flr,2)=0 then return frac else return 1-frac end \+ if;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Because " }{TEXT 0 1 "g" }{TEXT -1 66 " is defined as a Ma ple procedure, we must delay the evaluation of " }{TEXT 297 4 "g(t)" } {TEXT -1 13 " with quotes " }{TEXT 297 6 "'g(t)'" }{TEXT -1 8 " in the " }{TEXT 0 4 "plot" }{TEXT -1 10 " command. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "plot('g(t)' ,t=0..6.5,-0.2..1.2,thickness=2,numpoints=100,\n color=COLOR(RGB,. 9,.4,0),labels=[t,`f(t)`],ytickmarks=3);" }}{PARA 13 "" 1 "" {GLPLOT2D 611 160 160 {PLOTDATA 2 "6(-%'CURVESG6#7^s7$$\"\"!F)F(7$$\"3 CihhhhSpo!#>F+7$$\"3OBBBtRk%G\"!#=F/7$$\"3U5555R#o&>F1F37$$\"3E#>>>\\f Mj#F1F67$$\"3>$HHH9zoI$F1F97$$\"3@888j`AJRF1F<7$$\"3uggg5(*pxXF1F?7$$ \"3ubbb0WGY_F1FB7$$\"3&4444%\\s7fF1FE7$$\"3K[ZZZ4C)f'F1FH7$$\"30?>>>Q/ -sF1FK7$$\"3?WUUUGy\")yF1FN7$$\"3!*RRRRHJk&)F1FQ7$$\"3Gfdddr0A#*F1FT7$ $\"3z:99RXq?&*F1FW7$$\"3>rqq?>N>)*F1FZ7$$\"3VddKw?83**F1Fgn7$$\"3yWW%> B7p***F1Fjn7$$\"3@8jvQ#p&35!#<$\"3(y'oV7wI9**F17$$\"3%===VDZu,\"F_o$\" 3j\"==oXFb#)*F17$$\"3J>>W&G._.\"F_o$\"3#p!3eXr'zk*F17$$\"3yccc;$fH0\"F _o$\"3AKMMMoSq%*F17$$\"3LEEEmg786F_o$\"3mOPPP$R(o))F17$$\"3maaa*H9J=\" F_o$\"3Q`aa/q&)o\")F17$$\"3;000Xd1X7F_o$\"3W[\\\\\\DM\\vF17$$\"3waaa*H OIJ\"F_o$\"3W_aa/qjpoF17$$\"3I111T.wx8F_o$\"3.PRR*e'RAiF17$$\"3kQQQ)3$ HX9F_o$\"3o8;;;\"pqa&F17$$\"3:(oo=B4t]\"F_o$\"3^GJJ\"o2p#\\F17$$\"3#32 22--Ud\"F_o$\"3u\"HHHzzzD%F17$$\"3\\VVVG[oV;F_o$\"33lll:<:jNF17$$\"3.# ==o\")pTq\"F_o$\"3uz\"==$=IeHF17$$\"3iJJJh_\\pF_o$\"37PXXXNP*p*F-7$$\" 3CFFx%zY\\$>F_o$\"3!fFFF_?`]'F-7$$\"33+++Dt)o'>F_o$\"3[#******\\n7J$F- 7$$\"3aIIb\"e>Y)>F_o$\"3$o%ppW=/Q:F-7$$\"3*411\"Q=N-?F_o$\"3K))411\"Q= N#!#?7$$\"3+\"4fY4%3??F_o$\"33+\"4fY4%3?F-7$$\"3Y@@@^j\"y.#F_o$\"3uX@@ @^j\"y$F-7$$\"3geee$z#op?F_o$\"3Ygeee$z#opF-7$$\"3v&fffB\\:5#F_o$\"3_d fffB\\:5F17$$\"3Q%RRRp(fp@F_o$\"3)Q%RRRp(fp\"F17$$\"3)eee3rf7B#F_o$\"3 weee3rf7BF17$$\"3)GCCCes')H#F_o$\"3#)GCCCes')HF17$$\"3)eff4%G5iBF_o$\" 3%)eff4%G5i$F17$$\"3>tssPdSGCF_o$\"3%>tssPdSG%F17$$\"3UQQQQ*GK\\#F_o$ \"36%QQQQ*GK\\F17$$\"37(pp>#G4hDF_o$\"3?rpp>#G4h&F17$$\"3Gonn2QXEEF_o$ \"3#Gonn2QXE'F17$$\"31'eee&[H$p#F_o$\"3zgeee&[H$pF17$$\"3wTTTECefFF_o$ \"3_<99kU#ef(F17$$\"3tRRR>Q\\?GF_o$\"3F(RRR>Q\\?)F17$$\"3=gff\\^I!*GF_ o$\"3)=gff\\^I!*)F17$$\"3obbbN_u_HF_o$\"3\"obbbN_u_*F17$$\"3O..Gu()QpH F_o$\"3WLI!Gu()Qp*F17$$\"3-^]+8B.')HF_o$\"35500IJKg)*F17$$\"3%[UnB3aV* HF_o$\"3T[UnB3aV**F17$$\"3n)zH<&en-IF_o$\"3G8?q#[TK(**F17$$\"31s@4@w*4 ,$F_o$\"3Sz#y!*yB+*)*F17$$\"3*eaa/R>$>IF_o$\"32TXX&41o!)*F17$$\"3I\"44 4Pz60$F_o$\"3'p344H1#)[*F17$$\"3sOOO^$RI3$F_o$\"3%Gjjj[1'p\"*F17$$\"3] **)*)*y@z`JF_o$\"3-0555#y?Y)F17$$\"3F%RR*G*e]@$F_o$\"3Mdgg52T\\yF17$$ \"3a888Qsf%G$F_o$\"3akoo=w-arF17$$\"35)yyy&Q(zM$F_o$\"30>@@@9E?lF17$$ \"3T&[[)>zHCCC!\\wa%F17$$\"3Kvuu9y.6OF_o$\"3%oCDD&=i*)QF17$ $\"3)*fffWvzwOF_o$\"3B+//aX-KKF17$$\"3dEEEc`JUPF_o$\"3KMPPPk%od#F17$$ \"3X<<F17$$\"35cbbX&*HtQF_o$\"3))QWWWX+n7F17$$\" 3Qqpp%e[\"QRF_o$\"3AiHII:9&='F-7$$\"35BAskQFsRF_o$\"3y*oxx_8Ex#F-7$$\" 3QvuuW\"*R1SF_o$\"3k#QvuuW\"*R'Ffu7$$\"3mwvvS2HPSF_o$\"3qlwvvS2HPF-7$$ \"3/xwwOB=oSF_o$\"3J/xwwOB=oF-7$$\"30!**)*)HVYOTF_o$\"3W+**)*)HVYO\"F1 7$$\"38wvv!em=?%F_o$\"3Jhdd2em=?F17$$\"3+rqqS'4rE%F_o$\"3%*4222k4rEF17 $$\"3vqpppiDNVF_o$\"3a2(pppiDN$F17$$\"3Uyxx-=-)R%F_o$\"3M%yxx-=-)RF17$ $\"3a999%H-BY%F_o$\"3\\XTTTH-BYF17$$\"3Ebaa%fvK`%F_o$\"3c_XXXfvK`F17$$ \"3mkkk/h`(f%F_o$\"3sYYYY5OvfF17$$\"3c)yyG^aKm%F_o$\"3g&)yyG^aKmF17$$ \"3tEEET%)3IZF_o$\"3Jnii7W)3I(F17$$\"3/$>>p#=_\"z%F_o$\"3RI>>p#=_\"zF1 7$$\"3)pmmm#y+d[F_o$\"3spmmm#y+d)F17$$\"37tssU`*>#\\F_o$\"3EJFFFM&*>#* F17$$\"3Ma``jNvc\\F_o$\"3YVNNNc`n&*F17$$\"3cNMM%y6:*\\F_o$\"3lbVVVy6:* *F17$$\"3#)[[[3W=A]F_o$\"3%=^^^\"f:y(*F17$$\"3'HEEE.dG0&F_o$\"3Wqttt'H 9Z*F17$$\"3NWWW9njB^F_o$\"3]cbbbGjj()F17$$\"3YSSS?*4v=&F_o$\"3O&fffz+ \\7)F17$$\"3s///>\"42D&F_o$\"3%G&ff4)3H\\(F17$$\"3*4000co'=`F_o$\"34! \\\\\\R9L\"oF17$$\"3o#>>>\"H!pQ&F_o$\"3F_$)[J)=F-7$$\"3m++vQ?S*)fF_o$\"3QM***\\7'zf5F-7$$\"3UP 6I-)=N*fF_o$\"3/tD'))p(>\"['Ffu7$$\"3ItA&ecNw*fF_o$\"3e!pEx9MWO#Ffu7$$ \"3>4MSHBv,gF_o$\"3'=>4MSHBv\"Ffu7$$\"3'faaH4pe+'F_o$\"3[ifaaH4peFfu7$ $\"3F\"4fr9OB-'F_o$\"3'o74fr9OB#F-7$$\"3YPOO,K!)QgF_o$\"3GYPOO,K!)QF-7 $$\"3qaaa/GrtgF_o$\"3Aqaaa/GrtF-7$$\"3$GFFxSA'3hF_o$\"3GGFFxSA'3\"F17$ $\"3#RIIIyRE<'F_o$\"3HJkF_o$\"3%)\\PPP)>HJ%F17$$\"3++++++++lF_o$\"3++++++++]F1-% &COLORG6&%$RGBG$\"\"*!\"\"$\"\"%F_bmF(-%*THICKNESSG6#\"\"#-%+AXESLABEL SG6$%\"tG%%f(t)G-%*AXESTICKSG6$%(DEFAULTG\"\"$-%%VIEWG6$;F($\"#lF_bm;$ !\"#F_bm$\"#7F_bm" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 40 "Laplace tr ansforms of periodic functions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "Formulas for " } {XPPEDIT 18 0 "L*[f(t)];" "6#*&%\"LG\"\"\"7#-%\"fG6#%\"tGF%" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "f(t);" "6#-%\"fG6#%\"tG" }{TEXT -1 25 " is a periodic function. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 26 "Given a periodic function " } {XPPEDIT 18 0 "f(t);" "6#-%\"fG6#%\"tG" }{TEXT -1 13 " with period " } {TEXT 267 1 "p" }{TEXT -1 28 ", we construct the function:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f*`(t) = phi(t)-u[p](t)*f( t);" "6#/-%#f*G6#%\"tG,&-%$phiG6#F'\"\"\"*&-&%\"uG6#%\"pG6#F'F,-%\"fG6 #F'F,!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " } {XPPEDIT 18 0 "u[p](t);" "6#-&%\"uG6#%\"pG6#%\"tG" }{TEXT -1 39 " is t he unit step function defined by " }{XPPEDIT 18 0 "u[p](t);" "6#-&%\" uG6#%\"pG6#%\"tG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "PIECEWISE([0, t < \+ p],[1, p <= t]);" "6#-%*PIECEWISEG6$7$\"\"!2%\"tG%\"pG7$\"\"\"1F*F)" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "Then" }}{PARA 256 "" 0 " " {TEXT -1 2 " " }{XPPEDIT 18 0 "`f*`(t) = PIECEWISE([f(t), 0 <= t an d t < p],[0, p <= t]);" "6#/-%#f*G6#%\"tG-%*PIECEWISEG6$7$-%\"fG6#F'31 \"\"!F'2F'%\"pG7$F11F3F'" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 263 17 "_________________" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Thus " }{XPPEDIT 18 0 "`f*`(t);" "6#-%#f*G6#%\"tG" }{TEXT -1 41 " is the function which coincides with as " }{XPPEDIT 18 0 "f(t);" "6#-%\"fG6#%\"tG" }{TEXT -1 35 " over the \"first period\" from 0 to " }{TEXT 269 1 "p" }{TEXT -1 26 ", and is zero thereafter. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 16 "For example, if " }{XPPEDIT 18 0 "f(t);" "6#-%\"fG6#%\"tG" }{TEXT -1 23 " is the \"square wave\": " }}{PARA 0 " " 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(t) = PIECEWISE([1, 0 <= t and t < 1],[-1, 1 <= t and t < 2]);" "6 #/-%\"fG6#%\"tG-%*PIECEWISEG6$7$\"\"\"31\"\"!F'2F'F,7$,$F,!\"\"31F,F'2 F'\"\"#" }{TEXT -1 3 " , " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 5 " and " }{XPPEDIT 18 0 "f(t);" "6#-%\"fG6#%\"tG " }{TEXT -1 28 " is periodic with period 2. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "f := t -> (-1)^fl oor(t):\nplot(f(t),t=0..7.5,-1.2..1.2,thickness=2,\n color=r ed,labels=[t,``],ytickmarks=3);" }}{PARA 13 "" 1 "" {GLPLOT2D 481 157 157 {PLOTDATA 2 "6(-%'CURVESG6#7jr7$$\"\"!F)$\"\"\"F)7$$\"3$*****\\ily M;!#=F*7$$\"3'***\\P4c?dIF/F*7$$\"3s***\\Pfkol%F/F*7$$\"3%3+]i:FrE'F/F *7$$\"3!***\\(=UO(pyF/F*7$$\"3a**\\i!R[Eh)F/F*7$$\"3=**\\Pf.cb$*F/F*7$ $\"3E****\\P4(ya*F/F*7$$\"3M**\\i::=S(*F/F*7$$\"3$**\\(o/oLO)*F/F*7$$ \"3U***\\P4#\\K**F/F*7$$\"3g[7GQ(p0)**F/F*7$$\"3*)\\7GQZ'G+\"!#<$!\"\" F)7$$\"3#\\PMF]sw+\"FTFU7$$\"3&**\\(=n-[75FTFU7$$\"3'**\\7GQU40\"FTFU7 $$\"3)**\\P%)\\/%*3\"FTFU7$$\"3++vV['f*o6FTFU7$$\"3-+vV)z9&[7FTFU7$$\" 3,+voH[629FTFU7$$\"3/++DcWDq:FTFU7$$\"32+]7Gz%Rr\"FTFU7$$\"3=+v$fLI[z \"FTFU7$$\"33++vVFrv=FTFU7$$\"31+]7.+K;>FTFU7$$\"3#)****\\is#p&>FTFU7$ $\"3#**\\(=#*3Bx>FTFU7$$\"3!)**\\(=_Mv*>FTFU7$$\"3suoHH/h-?FTF*7$$\"3& )\\(=nL'o2?FTF*7$$\"3bC19WAw7?FTF*7$$\"3o*\\i::Qy,#FTF*7$$\"3'*\\iSm** )z-#FTF*7$$\"3y***\\7yT\"Q?FTF*7$$\"3$****\\(=qS;@FTF*7$$\"31++DcAn%># FTF*7$$\"3-+vVBp\"oL#FTF*7$$\"3/++D\"oSe]#FTF*7$$\"3#)*****\\(f-\\EFTF *7$$\"3t*\\P4^%e:GFTF*7$$\"3)*\\(=Ux+$*)GFTF*7$$\"3!)****\\Pq,jHFTF*7$ $\"3WVt_So7tHFTF*7$$\"3a(oaNkOK)HFTF*7$$\"3gf$o]a\"H))HFTF*7$$\"3?J?eY kM$*HFTF*7$$\"3!Gq&4[8S)*HFTF*7$$\"3&[P4'\\iX.IFTFU7$$\"3;iSmbenBIFTFU 7$$\"3#*\\(=&RMFTFU7$$\"3O+vo/h5(e $FTFU7$$\"3;+]7y!)HYPFTFU7$$\"3$)\\7GQf(*GQFTFU7$$\"3%**\\P%)z`;\"RFTF U7$$\"3?]Pfe%Rw%RFTFU7$$\"3/++v=^i$)RFTFU7$$\"3EJ&piKB\"))RFTFU7$$\"3[ i!*yL:i#*RFTFU7$$\"3q$f38u>r*RFTFU7$$\"3%\\7G)[zh,SFTF*7$$\"3R(=nQO91, %FTF*7$$\"3&)\\i!*y2h>SFTF*7$$\"3wuV)*3OgPSFTF*7$$\"3o*\\i!RkfbSFTF*7$ $\"3e\\(=nVFL8%FTF*7$$\"3Q+]PM%e5@%FTF*7$$\"3%)****\\()emrVFTF*7$$\"3A +](o%*)yGXFTF*7$$\"3'**\\Pfe83o%FTF*7$$\"3?]7.2B@lZFTF*7$$\"3c**\\7G5h \\[FTF*7$$\"3)*\\(o/.Hv)[FTF*7$$\"3^*\\7G.Za#\\FTF*7$$\"3GuV)R.1W%\\FT F*7$$\"31\\i:N]Oj\\FTF*7$$\"3)o=Ud`WG(\\FTF*7$$\"3sC\"Gj.CB)\\FTF*7$$ \"3i$4@myjq)\\FTF*7$$\"3aiS\"p`.=*\\FTF*7$$\"3cIq?(GVl*\\FTF*7$$\"3Z** **\\PIG,]FTFU7$$\"3C**\\PfTD#3&FTFU7$$\"3!****\\7GDK;&FTFU7$$\"3m*\\PM -p*4`FTFU7$$\"3)3+]Pp)RqaFTFU7$$\"3$**\\P4^]8i&FTFU7$$\"3K+voa'Q\"zdFT FU7$$\"3))\\(o/.si&eFTFU7$$\"3Y***\\iS0M$fFTFU7$$\"3/PMF!G$f`fFTFU7$$ \"3iuoHa6ytfFTFU7$$\"3'Qf384vQ)fFTFU7$$\"3?7.KG!pR*fFTFU7$$\"3#=b\"pFTF*7$$\"35+ DcE\"oE&pFTF*7$$\"3_u=<^DCrpFTF*7$$\"3%)\\7yvp\")*)pFTF*7$$\"3)HfL>egW *pFTF*7$$\"3/Pf3)=/\"**pFTF*7$$\"34\"GQUzZP+(FTFU7$$\"39D1R+9R3qFTFU7$ $\"3N7`p7'yw,(FTFU7$$\"3c*****\\#e'p-(FTFU7$$\"34](ozm#=1rFTFU7$$\"3i+ v$4^*R&=(FTFU7$$\"3Q+D1k2/PtFTFU7$$\"3++++++++vFTFU-%'COLOURG6&%$RGBG$ \"*++++\"!\")F(F(-%*AXESTICKSG6$%(DEFAULTG\"\"$-%+AXESLABELSG6$%\"tG%! G-%*THICKNESSG6#\"\"#-%%VIEWG6$;F($\"#vFV;$!#7FV$\"#7FV" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "`f*`(t)" "6#-%#f*G6 #%\"tG" }{TEXT -1 13 " is given by " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "`f*`(t) = PIECEWISE([1, 0 <= t and t < 1],[-1, 1 <= \+ t and t < 2],[0, 2 <= t]);" "6#/-%#f*G6#%\"tG-%*PIECEWISEG6%7$\"\"\"31 \"\"!F'2F'F,7$,$F,!\"\"31F,F'2F'\"\"#7$F/1F7F'" }{TEXT -1 3 " . " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "`f*` := t -> piecewise(t<1,1,t<2,-1,0):\nplot(`f*`(t),t=0..7.5,-1 .2..1.2,thickness=2,\n color=blue,labels=[t,``],ytickmarks=3 );" }}{PARA 13 "" 1 "" {GLPLOT2D 463 150 150 {PLOTDATA 2 "6(-%'CURVESG 6#7co7$$\"\"!F)$\"\"\"F)7$$\"3$*****\\ilyM;!#=F*7$$\"3'***\\P4c?dIF/F* 7$$\"3s***\\Pfkol%F/F*7$$\"3%3+]i:FrE'F/F*7$$\"3!***\\(=UO(pyF/F*7$$\" 3a**\\i!R[Eh)F/F*7$$\"3=**\\Pf.cb$*F/F*7$$\"3E****\\P4(ya*F/F*7$$\"3M* *\\i::=S(*F/F*7$$\"3$**\\(o/oLO)*F/F*7$$\"3U***\\P4#\\K**F/F*7$$\"3g[7 GQ(p0)**F/F*7$$\"3*)\\7GQZ'G+\"!#<$!\"\"F)7$$\"3#\\PMF]sw+\"FTFU7$$\"3 &**\\(=n-[75FTFU7$$\"3'**\\7GQU40\"FTFU7$$\"3)**\\P%)\\/%*3\"FTFU7$$\" 3++vV['f*o6FTFU7$$\"3-+vV)z9&[7FTFU7$$\"3,+voH[629FTFU7$$\"3/++DcWDq:F TFU7$$\"32+]7Gz%Rr\"FTFU7$$\"3=+v$fLI[z\"FTFU7$$\"33++vVFrv=FTFU7$$\"3 1+]7.+K;>FTFU7$$\"3#)****\\is#p&>FTFU7$$\"3#**\\(=#*3Bx>FTFU7$$\"3!)** \\(=_Mv*>FTFU7$$\"3suoHH/h-?FTF(7$$\"3&)\\(=nL'o2?FTF(7$$\"3bC19WAw7?F TF(7$$\"3o*\\i::Qy,#FTF(7$$\"3'*\\iSm**)z-#FTF(7$$\"3y***\\7yT\"Q?FTF( 7$$\"3$****\\(=qS;@FTF(7$$\"31++DcAn%>#FTF(7$$\"3-+vVBp\"oL#FTF(7$$\"3 /++D\"oSe]#FTF(7$$\"3#)*****\\(f-\\EFTF(7$$\"3t*\\P4^%e:GFTF(7$$\"3!)* ***\\Pq,jHFTF(7$$\"30+v$f)QxCJFTF(7$$\"33+D\"yl/)yKFTF(7$$\"3,+]P4'>&R MFTF(7$$\"3O+vo/h5(e$FTF(7$$\"3;+]7y!)HYPFTF(7$$\"3%**\\P%)z`;\"RFTF(7 $$\"3o*\\i!RkfbSFTF(7$$\"3Q+]PM%e5@%FTF(7$$\"3%)****\\()emrVFTF(7$$\"3 A+](o%*)yGXFTF(7$$\"3'**\\Pfe83o%FTF(7$$\"3c**\\7G5h\\[FTF(7$$\"3Z**** \\PIG,]FTF(7$$\"3!****\\7GDK;&FTF(7$$\"3m*\\PM-p*4`FTF(7$$\"3)3+]Pp)Rq aFTF(7$$\"3$**\\P4^]8i&FTF(7$$\"3K+voa'Q\"zdFTF(7$$\"3Y***\\iS0M$fFTF( 7$$\"33+vV)R3\\4'FTF(7$$\"3E+++v\\X]iFTF(7$$\"3^+]i:Q_4kFTF(7$$\"3W+v$ f[vsc'FTF(7$$\"3!3++v)GB7nFTF(7$$\"3m+]7G/PyoFTF(7$$\"3c*****\\#e'p-(F TF(7$$\"3i+v$4^*R&=(FTF(7$$\"3Q+D1k2/PtFTF(7$$\"3++++++++vFTF(-%'COLOU RG6&%$RGBGF(F($\"*++++\"!\")-%*AXESTICKSG6$%(DEFAULTG\"\"$-%+AXESLABEL SG6$%\"tG%!G-%*THICKNESSG6#\"\"#-%%VIEWG6$;F($\"#vFV;$!#7FV$\"#7FV" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Now, for a general periodic fun ction " }{XPPEDIT 18 0 "f(t);" "6#-%\"fG6#%\"tG" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[`f*`(t)] = L*[f(t) -u[p](t)*f(t)];" "6#/*&%\"LG\"\"\"7#-%#f*G6#%\"tGF&*&F%F&7#,&-%\"fG6#F +F&*&-&%\"uG6#%\"pG6#F+F&-F06#F+F&!\"\"F&" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "L*[f(t)]-L*[u[p](t)*f(t)];" "6#,&*&%\"LG\"\"\"7#-%\"fG6#%\"tGF&F &*&F%F&7#*&-&%\"uG6#%\"pG6#F+F&-F)6#F+F&F&!\"\"" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = L*[f(t)]-exp(-p* s)*L*[f(t+p)];" "6#/%!G,&*&%\"LG\"\"\"7#-%\"fG6#%\"tGF(F(*(-%$expG6#,$ *&%\"pGF(%\"sGF(!\"\"F(F'F(7#-F+6#,&F-F(F4F(F(F6" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 31 "using the second shift formula." }}{PARA 0 "" 0 "" {TEXT -1 12 "Then, since " }{XPPEDIT 18 0 "f(t);" "6#-%\"fG6 #%\"tG" }{TEXT -1 25 " is periodic with period " }{TEXT 270 1 "p" } {TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[ `f*`(t)] = L*[f(t)]-exp(-p*s)*L*[f(t)];" "6#/*&%\"LG\"\"\"7#-%#f*G6#% \"tGF&,&*&F%F&7#-%\"fG6#F+F&F&*(-%$expG6#,$*&%\"pGF&%\"sGF&!\"\"F&F%F& 7#-F06#F+F&F:" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = (1-exp(-p*s))*L *[f(t)];" "6#/%!G*(,&\"\"\"F'-%$expG6#,$*&%\"pGF'%\"sGF'!\"\"F/F'%\"LG F'7#-%\"fG6#%\"tGF'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 2 "so " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[f(t)];" "6#*&% \"LG\"\"\"7#-%\"fG6#%\"tGF%" }{TEXT -1 4 " = " }{XPPEDIT 18 0 "1/(1-e xp(-p*s));" "6#*&\"\"\"F$,&F$F$-%$expG6#,$*&%\"pGF$%\"sGF$!\"\"F-F-" } {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[`f*`(t)];" "6#*&%\"LG\"\"\"7#-%#f*G6 #%\"tGF%" }{TEXT -1 12 " ------- (i)" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 264 18 "__________________" }{TEXT -1 1 " " }{TEXT 258 12 " \+ " }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 1 ":" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[`f*`(t)] = Int(`f* `(t)*exp(-t*s),t = 0 .. infinity);" "6#/*&%\"LG\"\"\"7#-%#f*G6#%\"tGF& -%$IntG6$*&-F)6#F+F&-%$expG6#,$*&F+F&%\"sGF&!\"\"F&/F+;\"\"!%)infinity G" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 "= " }{XPPEDIT 18 0 "Int(`f*`(t)*exp(-t*s),t = 0 .. p);" "6#-%$IntG6$*&-%#f*G6#%\"tG\"\" \"-%$expG6#,$*&F*F+%\"sGF+!\"\"F+/F*;\"\"!%\"pG" }{TEXT -1 1 "," }} {PARA 0 "" 0 "" {TEXT -1 6 "since " }{XPPEDIT 18 0 "`f*`(t) = 0;" "6#/ -%#f*G6#%\"tG\"\"!" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "t>p" "6#2%\"p G%\"tG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "Thus" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[`f*`(t)] = Int(f(t)*exp( -t*s),t = 0 .. p);" "6#/*&%\"LG\"\"\"7#-%#f*G6#%\"tGF&-%$IntG6$*&-%\"f G6#F+F&-%$expG6#,$*&F+F&%\"sGF&!\"\"F&/F+;\"\"!%\"pG" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 32 "so the formula (i) implies that " }} {PARA 256 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "L*[f(t)];" "6#*&% \"LG\"\"\"7#-%\"fG6#%\"tGF%" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "1/(1-ex p(-p*s));" "6#*&\"\"\"F$,&F$F$-%$expG6#,$*&%\"pGF$%\"sGF$!\"\"F-F-" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(t)*exp(-t*s),t = 0 .. p);" "6#-%$ IntG6$*&-%\"fG6#%\"tG\"\"\"-%$expG6#,$*&F*F+%\"sGF+!\"\"F+/F*;\"\"!%\" pG" }{TEXT -1 16 " ------- (ii). " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{TEXT 265 21 "_____________________" }{TEXT -1 17 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "For \+ example, the formula (ii) can be used to find " }{XPPEDIT 18 0 "L*[sin *t];" "6#*&%\"LG\"\"\"7#*&%$sinGF%%\"tGF%F%" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[sin*t] = 1/(1-exp(-2*Pi* s));" "6#/*&%\"LG\"\"\"7#*&%$sinGF&%\"tGF&F&*&F&F&,&F&F&-%$expG6#,$*( \"\"#F&%#PiGF&%\"sGF&!\"\"F5F5" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(ex p(-t*s)*sin*t,t = 0 .. 2*Pi);" "6#-%$IntG6$*(-%$expG6#,$*&%\"tG\"\"\"% \"sGF-!\"\"F-%$sinGF-F,F-/F,;\"\"!*&\"\"#F-%#PiGF-" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 48 "Using integration by parts it can be show n that " }{XPPEDIT 18 0 "Int(exp(-t*s)*sin*t,t = 0 .. 2*Pi) = (1-exp(- 2*Pi*s))/(s^2+1);" "6#/-%$IntG6$*(-%$expG6#,$*&%\"tG\"\"\"%\"sGF.!\"\" F.%$sinGF.F-F./F-;\"\"!*&\"\"#F.%#PiGF.*&,&F.F.-F)6#,$*(F6F.F7F.F/F.F0 F0F.,&*$F/F6F.F.F.F0" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 " Hence " }{XPPEDIT 18 0 "L*[sin*t] = 1/(s^2+1);" "6#/*&%\"LG\"\"\"7#*&% $sinGF&%\"tGF&F&*&F&F&,&*$%\"sG\"\"#F&F&F&!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "Int(sin(t)*exp(-s*t),t=0..2*Pi)/(1-exp(-2*Pi*s));\n``=value(%);\n` `=simplify(expand(rhs(%)));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$ IntG6$*&-%$sinG6#%\"tG\"\"\"-%$expG6#,$*&%\"sGF,F+F,!\"\"F,/F+;\"\"!,$ *&\"\"#F,%#PiGF,F,*$,&F,F,-F.6#,$*(F9F,F:F,F2F,F3F3F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G**,&-%$expG6#,$*(\"\"#\"\"\"%#PiGF-%\"sGF-F-F- F-!\"\"F-,&*$)F/F,F-F-F-F-F0-F(6#,$*(F,F-F.F-F/F-F0F-,&F-F-F4F0F0" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&\"\"\"F&,&*$)%\"sG\"\"#F&F&F&F&! \"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 39 "Alternativ e derivation of formulas for " }{XPPEDIT 18 0 "L*[f(t)];" "6#*&%\"LG\" \"\"7#-%\"fG6#%\"tGF%" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "f(t);" " 6#-%\"fG6#%\"tG" }{TEXT -1 13 " is periodic." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 40 "An alternative w ay to obtain the formula" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[f(t)];" "6#*&%\"LG\"\"\"7#-% \"fG6#%\"tGF%" }{TEXT -1 4 " = " }{XPPEDIT 18 0 "1/(1-exp(-p*s));" "6 #*&\"\"\"F$,&F$F$-%$expG6#,$*&%\"pGF$%\"sGF$!\"\"F-F-" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "L*[`f*`(t)];" "6#*&%\"LG\"\"\"7#-%#f*G6#%\"tGF%" } {TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 15 "or equivalently" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[f(t)];" "6#*&%\"LG \"\"\"7#-%\"fG6#%\"tGF%" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "1/(1-exp(-p *s));" "6#*&\"\"\"F$,&F$F$-%$expG6#,$*&%\"pGF$%\"sGF$!\"\"F-F-" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(t)*exp(-t*s),t = 0 .. p);" "6#-%$ IntG6$*&-%\"fG6#%\"tG\"\"\"-%$expG6#,$*&F*F+%\"sGF+!\"\"F+/F*;\"\"!%\" pG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "for the Laplace transform of a periodic function " } {XPPEDIT 18 0 "f(t);" "6#-%\"fG6#%\"tG" }{TEXT -1 13 " with period " } {TEXT 268 1 "p" }{TEXT -1 15 " is as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[f( t)];" "6#*&%\"LG\"\"\"7#-%\"fG6#%\"tGF%" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "Int(f(t)*exp(-t*s),t = 0 .. infinity);" "6#-%$IntG6$*&-%\"fG6#% \"tG\"\"\"-%$expG6#,$*&F*F+%\"sGF+!\"\"F+/F*;\"\"!%)infinityG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(f (t)*exp(-t*s),t = 0 .. p)+Int(f(t)*exp(-t*s),t = p .. 2*p)+Int(f(t)*ex p(-t*s),t = 2*p .. 3*p)+` . . . `;" "6#/%!G,*-%$IntG6$*&-%\"fG6#%\"tG \"\"\"-%$expG6#,$*&F-F.%\"sGF.!\"\"F./F-;\"\"!%\"pGF.-F'6$*&-F+6#F-F.- F06#,$*&F-F.F4F.F5F./F-;F9*&\"\"#F.F9F.F.-F'6$*&-F+6#F-F.-F06#,$*&F-F. F4F.F5F./F-;*&FFF.F9F.*&\"\"$F.F9F.F.%(~.~.~.~GF." }{TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT -1 14 "\nSubstituting " }{XPPEDIT 18 0 "tau = t- p;" "6#/%$tauG,&%\"tG\"\"\"%\"pG!\"\"" }{TEXT -1 22 " in the 2nd integ ral, " }{XPPEDIT 18 0 "tau = t-2*p;" "6#/%$tauG,&%\"tG\"\"\"*&\"\"#F'% \"pGF'!\"\"" }{TEXT -1 40 " in the 3rd integral, and so on, gives: " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "L*[f(t)] = Int(f(t)*exp(-t*s),t = 0 .. p)+Int(f(tau+p)* exp(-(tau+p)*s),tau = 0 .. p)+Int(f(tau+2*p)*exp(-(tau+2*p)*s),tau = 0 .. p)+` . . . `;" "6#/*&%\"LG\"\"\"7#-%\"fG6#%\"tGF&,*-%$IntG6$*&-F)6 #F+F&-%$expG6#,$*&F+F&%\"sGF&!\"\"F&/F+;\"\"!%\"pGF&-F.6$*&-F)6#,&%$ta uGF&F=F&F&-F46#,$*&,&FDF&F=F&F&F8F&F9F&/FD;FF/ F1F/F2F//F>;F5F0F/" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 14 "Since the sum " }{XPPEDIT 18 0 "Sum(exp(- k*p*s),k = 0 .. infinity);" "6#-%$SumG6$-%$expG6#,$*(%\"kG\"\"\"%\"pGF ,%\"sGF,!\"\"/F+;\"\"!%)infinityG" }{TEXT -1 46 " is an infinite geome tric series of the form " }{XPPEDIT 18 0 "1+r+r^2;" "6#,(\"\"\"F$%\"r GF$*$F%\"\"#F$" }{TEXT -1 12 ". . , where " }{XPPEDIT 18 0 "r = exp(-p *s);" "6#/%\"rG-%$expG6#,$*&%\"pG\"\"\"%\"sGF+!\"\"" }{TEXT -1 27 ", i t converges to the sum " }{XPPEDIT 18 0 "1/(1-r);" "6#*&\"\"\"F$,&F$F $%\"rG!\"\"F'" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "1/(1-exp(-p*s));" "6# *&\"\"\"F$,&F$F$-%$expG6#,$*&%\"pGF$%\"sGF$!\"\"F-F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "Thus" }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "L*[f(t)] = 1/(1-exp(-p*s));" "6#/*&%\"LG\"\"\"7#-%\" fG6#%\"tGF&*&F&F&,&F&F&-%$expG6#,$*&%\"pGF&%\"sGF&!\"\"F5F5" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(t)*exp(-t*s),t = 0 .. p);" "6#-%$IntG6$ *&-%\"fG6#%\"tG\"\"\"-%$expG6#,$*&F*F+%\"sGF+!\"\"F+/F*;\"\"!%\"pG" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 266 22 "_____ _________________" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 62 "Examples of Laplace transforms o f piecewise periodic functions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 58 "The following examples demonst rate the use of the formula:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "L*[f(t)];" "6#*&%\"LG\"\"\"7#-%\"fG6#%\"tGF%" }{TEXT -1 4 " = " }{XPPEDIT 18 0 "1/(1-exp(-p*s));" "6#*&\"\"\"F$,&F$F$-%$ex pG6#,$*&%\"pGF$%\"sGF$!\"\"F-F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "L*[`f *`(t)]" "6#*&%\"LG\"\"\"7#-%#f*G6#%\"tGF%" }{TEXT -1 13 " ------- (i) \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 286 18 "__________________ " }{TEXT -1 17 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 71 "for calculating the Laplace transform of \+ a piecewise periodic function " }{XPPEDIT 18 0 "f(t);" "6#-%\"fG6#%\"t G" }{TEXT -1 13 " with period " }{TEXT 284 1 "p" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 1 ":" }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f*`(t)" "6#-%#f*G6#%\"tG" }{TEXT -1 28 " is the function defined by " }{XPPEDIT 18 0 "`f*`(t) = PIECEWISE( [f(t), 0 <= t and t < p],[0, p <= t]);" "6#/-%#f*G6#%\"tG-%*PIECEWISEG 6$7$-%\"fG6#F'31\"\"!F'2F'%\"pG7$F11F3F'" }{TEXT -1 1 "," }}{PARA 15 " " 0 "" {TEXT -1 1 " " }{TEXT 285 1 "p" }{TEXT -1 15 " is the period." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 45 "Example 1: Laplace tra nsform of a square wave" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(t) = PIECEWISE([1, 0 <= t and \+ t < 1],[-1, 1 <= t and t < 2]);" "6#/-%\"fG6#%\"tG-%*PIECEWISEG6$7$\" \"\"31\"\"!F'2F'F,7$,$F,!\"\"31F,F'2F'\"\"#" }{TEXT -1 3 " , " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 " and " } {XPPEDIT 18 0 "f(t);" "6#-%\"fG6#%\"tG" }{TEXT -1 28 " is periodic wit h period 2. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "plot((-1)^floor(t),t=0..7.5,-1.2..1.2,thickness=2 ,\n color=COLOR(RGB,0,.5,.2),labels=[t,`f(t)`],ytickmarks=3) ;" }}{PARA 13 "" 1 "" {GLPLOT2D 538 179 179 {PLOTDATA 2 "6(-%'CURVESG6 #7jr7$$\"\"!F)$\"\"\"F)7$$\"3$*****\\ilyM;!#=F*7$$\"3'***\\P4c?dIF/F*7 $$\"3s***\\Pfkol%F/F*7$$\"3%3+]i:FrE'F/F*7$$\"3!***\\(=UO(pyF/F*7$$\"3 a**\\i!R[Eh)F/F*7$$\"3=**\\Pf.cb$*F/F*7$$\"3E****\\P4(ya*F/F*7$$\"3M** \\i::=S(*F/F*7$$\"3$**\\(o/oLO)*F/F*7$$\"3U***\\P4#\\K**F/F*7$$\"3g[7G Q(p0)**F/F*7$$\"3*)\\7GQZ'G+\"!#<$!\"\"F)7$$\"3#\\PMF]sw+\"FTFU7$$\"3& **\\(=n-[75FTFU7$$\"3'**\\7GQU40\"FTFU7$$\"3)**\\P%)\\/%*3\"FTFU7$$\"3 ++vV['f*o6FTFU7$$\"3-+vV)z9&[7FTFU7$$\"3,+voH[629FTFU7$$\"3/++DcWDq:FT FU7$$\"32+]7Gz%Rr\"FTFU7$$\"3=+v$fLI[z\"FTFU7$$\"33++vVFrv=FTFU7$$\"31 +]7.+K;>FTFU7$$\"3#)****\\is#p&>FTFU7$$\"3#**\\(=#*3Bx>FTFU7$$\"3!)** \\(=_Mv*>FTFU7$$\"3suoHH/h-?FTF*7$$\"3&)\\(=nL'o2?FTF*7$$\"3bC19WAw7?F TF*7$$\"3o*\\i::Qy,#FTF*7$$\"3'*\\iSm**)z-#FTF*7$$\"3y***\\7yT\"Q?FTF* 7$$\"3$****\\(=qS;@FTF*7$$\"31++DcAn%>#FTF*7$$\"3-+vVBp\"oL#FTF*7$$\"3 /++D\"oSe]#FTF*7$$\"3#)*****\\(f-\\EFTF*7$$\"3t*\\P4^%e:GFTF*7$$\"3)* \\(=Ux+$*)GFTF*7$$\"3!)****\\Pq,jHFTF*7$$\"3WVt_So7tHFTF*7$$\"3a(oaNkO K)HFTF*7$$\"3gf$o]a\"H))HFTF*7$$\"3?J?eYkM$*HFTF*7$$\"3!Gq&4[8S)*HFTF* 7$$\"3&[P4'\\iX.IFTFU7$$\"3;iSmbenBIFTFU7$$\"3#*\\(=&RMFTFU7$$\"3O+vo/h5(e$FTFU7$$\"3;+]7y!)HYPFTFU7$$\" 3$)\\7GQf(*GQFTFU7$$\"3%**\\P%)z`;\"RFTFU7$$\"3?]Pfe%Rw%RFTFU7$$\"3/++ v=^i$)RFTFU7$$\"3EJ&piKB\"))RFTFU7$$\"3[i!*yL:i#*RFTFU7$$\"3q$f38u>r*R FTFU7$$\"3%\\7G)[zh,SFTF*7$$\"3R(=nQO91,%FTF*7$$\"3&)\\i!*y2h>SFTF*7$$ \"3wuV)*3OgPSFTF*7$$\"3o*\\i!RkfbSFTF*7$$\"3e\\(=nVFL8%FTF*7$$\"3Q+]PM %e5@%FTF*7$$\"3%)****\\()emrVFTF*7$$\"3A+](o%*)yGXFTF*7$$\"3'**\\Pfe83 o%FTF*7$$\"3?]7.2B@lZFTF*7$$\"3c**\\7G5h\\[FTF*7$$\"3)*\\(o/.Hv)[FTF*7 $$\"3^*\\7G.Za#\\FTF*7$$\"3GuV)R.1W%\\FTF*7$$\"31\\i:N]Oj\\FTF*7$$\"3) o=Ud`WG(\\FTF*7$$\"3sC\"Gj.CB)\\FTF*7$$\"3i$4@myjq)\\FTF*7$$\"3aiS\"p` .=*\\FTF*7$$\"3cIq?(GVl*\\FTF*7$$\"3Z****\\PIG,]FTFU7$$\"3C**\\PfTD#3& FTFU7$$\"3!****\\7GDK;&FTFU7$$\"3m*\\PM-p*4`FTFU7$$\"3)3+]Pp)RqaFTFU7$ $\"3$**\\P4^]8i&FTFU7$$\"3K+voa'Q\"zdFTFU7$$\"3))\\(o/.si&eFTFU7$$\"3Y ***\\iS0M$fFTFU7$$\"3/PMF!G$f`fFTFU7$$\"3iuoHa6ytfFTFU7$$\"3'Qf384vQ)f FTFU7$$\"3?7.KG!pR*fFTFU7$$\"3#=b\"pFTF*7$$\"35+DcE\"oE&pFTF*7$$\"3_u=<^DCrpFTF* 7$$\"3%)\\7yvp\")*)pFTF*7$$\"3)HfL>egW*pFTF*7$$\"3/Pf3)=/\"**pFTF*7$$ \"34\"GQUzZP+(FTFU7$$\"39D1R+9R3qFTFU7$$\"3N7`p7'yw,(FTFU7$$\"3c***** \\#e'p-(FTFU7$$\"34](ozm#=1rFTFU7$$\"3i+v$4^*R&=(FTFU7$$\"3Q+D1k2/PtFT FU7$$\"3++++++++vFTFU-%&COLORG6&%$RGBGF)$\"\"&FV$\"\"#FV-%*AXESTICKSG6 $%(DEFAULTG\"\"$-%+AXESLABELSG6$%\"tG%%f(t)G-%*THICKNESSG6#Fhcl-%%VIEW G6$;F($\"#vFV;$!#7FV$\"#7FV" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 41.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "The associated (non-periodic) function which coincides wi th " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 52 " over the \+ \"first period\" and is zero thereafter is: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f*`(t) = P IECEWISE([1, 0 <= t and t < 1],[-1, 1 <= t and t < 2],[0, 2 <= t]);" " 6#/-%#f*G6#%\"tG-%*PIECEWISEG6%7$\"\"\"31\"\"!F'2F'F,7$,$F,!\"\"31F,F' 2F'\"\"#7$F/1F7F'" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1+u[1](t)*(-2)+u[2 ](t)" "6#/%!G,(\"\"\"F&*&-&%\"uG6#F&6#%\"tGF&,$\"\"#!\"\"F&F&-&F*6#F/6 #F-F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 1 "u" }{TEXT -1 4 " .. " } {HYPERLNK 17 "u" 1 "" "u" }{TEXT -1 88 " from the subsection which fol lows the examples, and which implements the step function " }{XPPEDIT 18 0 "u[a](t)=`` " "6#/-&%\"uG6#%\"aG6#%\"tG%!G" }{TEXT 297 7 "u[a](t) " }{TEXT -1 42 " may be used to set up the Maple function " }{TEXT 0 4 "`f*`" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "`f*`(t)" "6#-%#f*G6#%\"tG " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 198 "`f*` := t -> 1+'u[1]'(t)*(-2)+'u[2]'(t):\n'` f*`(t)'=`f*`(t);\n``=convert(rhs(%),piecewise);\nplot(`f*`(t),t=0..5.5 ,-1.2..1.2,thickness=2,color=COLOR(RGB,0,.2,.8),\n labels=[t,`f*(t )`],ytickmarks=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f*G6#%\"tG,( \"\"\"F)*&\"\"#F)-&%\"uG6#F)F&F)!\"\"-&F.6#F+F&F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%*PIECEWISEG6%7$\"\"\"2%\"tGF)7$!\"\"2F+\"\"#7$\" \"!1F/F+" }}{PARA 13 "" 1 "" {GLPLOT2D 528 205 205 {PLOTDATA 2 "6(-%'C URVESG6#7co7$$\"\"!F)$\"\"\"F)7$$\"+zM%))>\"!#5F*7$$\"+!y]>C#F/F*7$$\" +MS.:MF/F*7$$\"+ZK*ef%F/F*7$$\"+T+9rdF/F*7$$\"+iUugoF/F*7$$\"+?j'*))zF /F*7$$\"+b2Ps&)F/F*7$$\"+'=vd:*F/F*7$$\"+5>aY%*F/F*7$$\"+N'3tt*F/F*7$$ \"+?.+5)*F/F*7$$\"++?p#))*F/F*7$$\"+Sy.>**F/F*7$$\"+!o$Qb**F/F*7$$\"+? &H<***F/F*7$$\"+Ov!G+\"!\"*$!\"\"F)7$$\"+seM<5FjnF[o7$$\"+3U)=.\"FjnF[ o7$$\"+r?q\"4\"FjnF[o7$$\"+M*>::\"FjnF[o7$$\"+Z^*oD\"FjnF[o7$$\"+yE_v8 FjnF[o7$$\"+1tj%\\\"FjnF[o7$$\"+@jU4;FjnF[o7$$\"+Vdm8E%>FjnF[o7$$\"+9p)y&>FjnF[o7$$\"+! yaJ(>FjnF[o7$$\"+8()y!)>FjnF[o7$$\"+YEU))>FjnF[o7$$\"+7'RA*>FjnF[o7$$ \"+yl0'*>FjnF[o7$$\"+WN()**>FjnF[o7$$\"+60p.?FjnF(7$$\"+UiAM?FjnF(7$$ \"+u>wk?FjnF(7$$\"+n0#)=@FjnF(7$$\"+g\"zG<#FjnF(7$$\"+\\3]\"H#FjnF(7$$ \"+[nX/CFjnF(7$$\"+zVJADFjnF(7$$\"+wWaIEFjnF(7$$\"+c_GZFFjnF(7$$\"+=ha oGFjnF(7$$\"+aS5uHFjnF(7$$\"+=&4\")3$FjnF(7$$\"+<$))e?$FjnF(7$$\"+$*=6 @LFjnF(7$$\"+HmfKMFjnF(7$$\"+?9QcNFjnF(7$$\"+gvgnOFjnF(7$$\"+0_O'y$Fjn F(7$$\"+$GxR*QFjnF(7$$\"+3di6SFjnF(7$$\"+2PKATFjnF(7$$\"+7].QUFjnF(7$$ \"+jR;^VFjnF(7$$\"+\"\\*fpWFjnF(7$$\"+!)pm$e%FjnF(7$$\"+juJ+ZFjnF(7$$ \"+A?+;[FjnF(7$$\"+;TIA\\FjnF(7$$\"+7$QT/&FjnF(7$$\"+q#3J:&FjnF(7$$\"+ tHHp_FjnF(7$$\"+fl\\!Q&FjnF(7$$\"#bF\\oF(-%+AXESLABELSG6$%\"tG%&f*(t)G -%&COLORG6&%$RGBGF($\"\"#F\\o$\"\")F\\o-%*THICKNESSG6#Ffy-%*AXESTICKSG 6$%(DEFAULTG\"\"$-%%VIEWG6$;F(Fjx;$!#7F\\o$\"#7F\\o" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "The Laplace transform of " }{XPPEDIT 18 0 "`f*`(t )" "6#-%#f*G6#%\"tG" }{TEXT -1 46 " can be obtained by making use of t he result: " }{XPPEDIT 18 0 "L*[u[a](t)]=exp(-a*s)/s" "6#/*&%\"LG\"\" \"7#-&%\"uG6#%\"aG6#%\"tGF&*&-%$expG6#,$*&F,F&%\"sGF&!\"\"F&F5F6" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 298 1 " " }{XPPEDIT 18 0 "L*[`f*`(t)] = L*[1-2*u[1](t)+u[2](t)]; " "6#/*&%\"LG\"\"\"7#-%#f*G6#%\"tGF&*&F%F&7#,(F&F&*&\"\"#F&-&%\"uG6#F& 6#F+F&!\"\"-&F36#F06#F+F&F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/s-2* exp(-s)/s+exp(-2*s)/s;" "6#/%!G,(*&\"\"\"F'%\"sG!\"\"F'*(\"\"#F'-%$exp G6#,$F(F)F'F(F)F)*&-F-6#,$*&F+F'F(F'F)F'F(F)F'" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(1-2*exp(-s)+exp(- 2*s))/s" "6#/%!G*&,(\"\"\"F'*&\"\"#F'-%$expG6#,$%\"sG!\"\"F'F/-F+6#,$* &F)F'F.F'F/F'F'F.F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6 #%\"tG" }{TEXT -1 42 " has period 2, it follows using (i) that: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[f(t)];" "6#*&%\"LG \"\"\"7#-%\"fG6#%\"tGF%" }{TEXT -1 4 " = " }{XPPEDIT 18 0 "1/(1-exp(- 2*s));" "6#*&\"\"\"F$,&F$F$-%$expG6#,$*&\"\"#F$%\"sGF$!\"\"F-F-" } {TEXT -1 1 " " }{XPPEDIT 18 0 "``((1-2*exp(-s)+exp(-2*s))/s);" "6#-%!G 6#*&,(\"\"\"F(*&\"\"#F(-%$expG6#,$%\"sG!\"\"F(F0-F,6#,$*&F*F(F/F(F0F(F (F/F0" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (1-2*exp(-s)+exp(-2*s))/(s*(1-exp(-2*s)));" "6#/%!G*&,(\"\" \"F'*&\"\"#F'-%$expG6#,$%\"sG!\"\"F'F/-F+6#,$*&F)F'F.F'F/F'F'*&F.F',&F 'F'-F+6#,$*&F)F'F.F'F/F/F'F/" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = (1-exp(-s))^2/(s*(1-exp(-s))*(1+e xp(-s)));" "6#/%!G*&,&\"\"\"F'-%$expG6#,$%\"sG!\"\"F-\"\"#*(F,F',&F'F' -F)6#,$F,F-F-F',&F'F'-F)6#,$F,F-F'F'F-" }{TEXT -1 1 " " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (1-exp(-s))/(s*(1+exp(-s))) ;" "6#/%!G*&,&\"\"\"F'-%$expG6#,$%\"sG!\"\"F-F'*&F,F',&F'F'-F)6#,$F,F- F'F'F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "The Laplace transform of " }{XPPEDIT 18 0 "`f*`(t)" "6#-%#f*G6#%\"tG" }{TEXT -1 25 " can be determined using " }{TEXT 0 7 "laplace" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 212 "`f*` := t -> 1+'u[1]'(t)*(-2)+'u[2 ]'(t):\n'`f*`(t)'=`f*`(t);\nconvert(rhs(%),Heaviside):\n`f*` := unappl y(%,t):\n'`f*`(t)'=`f*`(t);\n`Laplace transform`=inttrans[laplace](`f* `(t),t,s);\n``=normal(rhs(%));\nFs := rhs(%):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f*G6#%\"tG,(\"\"\"F)*&\"\"#F)-&%\"uG6#F)F&F)!\"\"-& F.6#F+F&F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f*G6#%\"tG,(\"\"\"F) *&\"\"#F)-%*HeavisideG6#,&F)!\"\"F'F)F)F0-F-6#,&F+F0F'F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG,(*&\"\"\"F'%\"sG!\"\"F '*(\"\"#F'-%$expG6#,$F(F)F'F(F)F)*&-F-6#,$*&F+F'F(F'F)F'F(F)F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&,(-%$expG6#,$*&\"\"#\"\"\"%\"sGF -!\"\"F-*&F,F--F(6#,$F.F/F-F/F-F-F-F.F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Since the periodic function " } {XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 40 " has period 2, th e Laplace transform of " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" } {TEXT -1 42 " is obtained by dividing the transform of " }{XPPEDIT 18 0 "`f*`(t)" "6#-%#f*G6#%\"tG" }{TEXT -1 3 " by" }{XPPEDIT 18 0 " ``(1- exp(-2*s))" "6#-%!G6#,&\"\"\"F'-%$expG6#,$*&\"\"#F'%\"sGF'!\"\"F/" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Fs/(1-exp(-2*s));\n``=simplify(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#*(,(\"\"\"F%*&\"\"#F%-%$expG6#,$%\"sG!\"\"F%F--F )6#,$*&F'F%F,F%F-F%F%F,F-,&F%F%F.F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*(,&-%$expG6#,$%\"sG!\"\"\"\"\"F.F-F.,&F(F.F.F.F-F,F-F-" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Alternatively, we can start wit h the piecewise definition of " }{XPPEDIT 18 0 "`f*`(t)" "6#-%#f*G6#% \"tG" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f*`(t) = PIECEWISE([1, t < 1],[-1, 1 <= t and t < 2],[0, 2 <= t ]);" "6#/-%#f*G6#%\"tG-%*PIECEWISEG6%7$\"\"\"2F'F,7$,$F,!\"\"31F,F'2F' \"\"#7$\"\"!1F4F'" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 227 "`f*` := t -> piecewise(t<1 ,1,t<2,-1,t>=2,0):\n'`f*`(t)'=`f*`(t);\nsimplify(convert(`f*`(t),Heavi side)):\n`f*` := unapply(%,t):\n'`f*`(t)'=`f*`(t);\n`Laplace transform `=inttrans[laplace](`f*`(t),t,s);\n``=normal(rhs(%));\nFs := rhs(%):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f*G6#%\"tG-%*PIECEWISEG6%7$\"\" \"2F'F,7$!\"\"2F'\"\"#7$\"\"!1F1F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%#f*G6#%\"tG,(\"\"\"F)*&\"\"#F)-%*HeavisideG6#,&F)!\"\"F'F)F)F0-F-6# ,&F+F0F'F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG, (*&\"\"\"F'%\"sG!\"\"F'*(\"\"#F'-%$expG6#,$F(F)F'F(F)F)*&-F-6#,$*&F+F' F(F'F)F'F(F)F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&,(-%$expG6#,$* &\"\"#\"\"\"%\"sGF-!\"\"F-*&F,F--F(6#,$F.F/F-F/F-F-F-F.F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 " As before the La place transform of " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 42 " is obtained by dividing the transform of " }{XPPEDIT 18 0 "`f* `(t)" "6#-%#f*G6#%\"tG" }{TEXT -1 3 " by" }{XPPEDIT 18 0 " ``(1-exp(-2 *s))" "6#-%!G6#,&\"\"\"F'-%$expG6#,$*&\"\"#F'%\"sGF'!\"\"F/" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Fs/(1-exp(-2*s));\n``=simplify(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#*(,(\"\"\"F%*&\"\"#F%-%$expG6#,$%\"sG!\"\"F%F--F)6#,$ *&F'F%F,F%F-F%F%F,F-,&F%F%F.F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% !G,$*(,&-%$expG6#,$%\"sG!\"\"\"\"\"F.F-F.,&F(F.F.F.F-F,F-F-" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "We can also obtain this result \+ by using the integral formula:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[f(t)];" "6#*&%\"LG \"\"\"7#-%\"fG6#%\"tGF%" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "1/(1-exp(-p *s));" "6#*&\"\"\"F$,&F$F$-%$expG6#,$*&%\"pGF$%\"sGF$!\"\"F-F-" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(t)*exp(-t*s),t = 0 .. p);" "6#-%$ IntG6$*&-%\"fG6#%\"tG\"\"\"-%$expG6#,$*&F*F+%\"sGF+!\"\"F+/F*;\"\"!%\" pG" }{TEXT -1 3 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "f(t) = (-1)^floor(t);" "6#/-%\"fG6#%\"tG),$\"\"\"!\"\"-%&floorG6 #F'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "p = 2" "6#/%\"pG\"\"#" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "f := t -> (-1)^floor(t):\n'f(t)'=f(t);\nInt('f( t)'*exp(-s*t),t=0..2)/(1-exp(-2*s));\n``=value(%);\n``=simplify(map(ex pand,rhs(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"tG)!\"\" -%&floorGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$IntG6$*&-%\"fG6#% \"tG\"\"\"-%$expG6#,$*&%\"sGF,F+F,!\"\"F,/F+;\"\"!\"\"#*$,&F,F,-F.6#,$ *&F7F,F2F,F3F3F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$**,(-%$expG6 #,$*&\"\"#\"\"\"%\"sGF.F.!\"\"F.F0*&F-F.-F)6#F/F.F.F.-F)6#,$*&F-F.F/F. F0F.F/F0,&F.F.F4F0F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*(,&-%$e xpG6#%\"sG\"\"\"F+!\"\"F+,&F'F+F+F+F,F*F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Example 2: Laplace transform of a sawtoot h wave" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(t) =t" "6#/-%\"fG6#%\"tGF'" }{TEXT -1 5 ", \+ " }{XPPEDIT 18 0 "0<=t" "6#1\"\"!%\"tG" }{XPPEDIT 18 0 "``<1" "6#2%!G \"\"\"" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 5 " and " } {XPPEDIT 18 0 "f(t);" "6#-%\"fG6#%\"tG" }{TEXT -1 28 " is periodic wit h period 1. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "plot(t-floor(t),t=0..5.5,-0.2..1.2,thickness=2,nu mpoints=100,\n color=COLOR(RGB,.6,.6,0),labels=[t,`f(t)`],ytickmar ks=3);" }}{PARA 13 "" 1 "" {GLPLOT2D 593 185 185 {PLOTDATA 2 "6(-%'CUR VESG6#7gt7$$\"\"!F)F(7$$\"3'RWWWWuD\"e!#>F+7$$\"3#*))))))Qk+(3\"!#=F/7 $$\"3oxxxxSxb;F1F37$$\"3txxxx=JGAF1F67$$\"3Ybbb0&G\")z#F1F97$$\"3#4666 Y@kK$F1F<7$$\"3Wmmm;#QM(QF1F?7$$\"3NxxxFP;RWF1FB7$$\"3N******\\\\2.]F1 FE7$$\"3;bbbbp7$e&F1FH7$$\"3Yxxxxq.%4'F1FK7$$\"3!pmmm'3?pmF1FN7$$\"3;L LLLjsYsF1FQ7$$\"3!HLLL8zK!yF1FT7$$\"3MWWW%R#o3$)F1FW7$$\"3cbbbbdl4*)F1 FZ7$$\"3k)))))))Q2U;*F1Fgn7$$\"3s@AAA!f(=%*F1Fjn7$$\"3Amm;a5\"oc*F1F]o 7$$\"3g466'3j[r*F1F`o7$$\"3IJL3-\"*)))y*F1Fco7$$\"35ab0=^\"H')*F1Ffo7$ $\"31m;/E\"G***)*F1Fio7$$\"3!pxFS8Tp$**F1F\\p7$$\"3KK3-QwWb**F1F_p7$$ \"3u()Q,UT&R(**F1Fbp7$$\"3;Vp+Y1Y#***F1Fep7$$\"3')*****\\r'4,5!#<$\"3M f)*****\\r'4\"!#?7$$\"3FWW%>%pIF5Fjp$\"3SFWW%>%pIFF-7$$\"3o))))))or^`5 Fjp$\"3))o))))))or^`F-7$$\"31+++:2.66Fjp$\"3e+++]rI56F17$$\"3gmmm6szl6 Fjp$\"35mmm;@(zl\"F17$$\"3mbbb0.%HA\"Fjp$\"3jcbbbISHAF17$$\"3!))))))Q] :aF\"Fjp$\"3-))))))Q]:aFF17$$\"3FWWW%4W\"Fjp$\"3P******\\%) *)>WF17$$\"3/666@_E(\\\"Fjp$\"3W5666Als\\F17$$\"35LLL$4qVb\"Fjp$\"3\"4 LLL$4qVbF17$$\"3!)******pgB5;Fjp$\"3(z******pgB5'F17$$\"3y*****\\F*Gk; 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 1 "u" }{TEXT -1 4 " .. " }{HYPERLNK 17 "u" 1 "" "u" }{TEXT -1 88 " from the subsect ion which follows the examples, and which implements the step function " }{XPPEDIT 18 0 "u[a](t)=`` " "6#/-&%\"uG6#%\"aG6#%\"tG%!G" }{TEXT 297 7 "u[a](t)" }{TEXT -1 42 " may be used to set up the Maple functio n " }{TEXT 0 4 "`f*`" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "`f*`(t)" "6# -%#f*G6#%\"tG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "`f*` := t -> t+'u[1]'(t)*(- t):\n'`f*`(t)'=`f*`(t);\n``=convert(rhs(%),piecewise);\nplot(`f*`(t),t =0..4.5,0..1.2,thickness=2,color=COLOR(RGB,.9,.5,.2),\n labels=[t, `f*(t)`],ytickmarks=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f*G6#% \"tG,&F'\"\"\"*&-&%\"uG6#F)F&F)F'F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%*PIECEWISEG6$7$%\"tG2F)\"\"\"7$\"\"!1F+F)" }}{PARA 13 "" 1 "" {GLPLOT2D 557 145 145 {PLOTDATA 2 "6(-%'CURVESG6#7ao7$$\"\"!F)F(7 $$\"+)ofV!\\!#6F+7$$\"+v$>(3)*F-F/7$$\"+_wf29!#5F27$$\"+mLKM=F4F67$$\" +h5A9BF4F97$$\"+c(=Tz#F4F<7$$\"+Dv>xKF4F?7$$\"+%Hw-w$F4FB7$$\"+u!f5C%F 4FE7$$\"+`=%=s%F4FH7$$\"+N!*en^F4FK7$$\"+;iL8cF4FN7$$\"+0;)[2'F4FQ7$$ \"+\"*pUOlF4FT7$$\"+!*yv8qF4FW7$$\"+\"z)3\"\\(F4FZ7$$\"+&))))o'zF4Fgn7 $$\"+y*)oU%)F4Fjn7$$\"+gy5K*)F4F]o7$$\"+Qn_@%*F4F`o7$$\"+Xp1P'*F4Fco7$ $\"+brg_)*F4Ffo7$$\"+5A\\1**F4Fio7$$\"+gsPg**F4F\\p7$$\"+&y>t)**F4F_p7 $$\"+JiU,5!\"*F(7$$\"+%[?T+\"FdpF(7$$\"+OZ\"o+\"FdpF(7$$\"+YFdpF(7 $$\"+m$[nMFdpF(7$$\"+WK/gNF dpF(7$$\"+R]%pl$FdpF(7$$\"+&)HF]PFdpF(7$$\"+*G9d%QFdpF(7$$\"+#Hl.%RFdp F(7$$\"+K(Rt-%FdpF(7$$\"+dA-FTFdpF(7$$\"+&\\zh@%FdpF(7$$\"+2(R7J%FdpF( 7$$\"+eWA-WFdpF(7$$\"#X!\"\"F(-%+AXESLABELSG6$%\"tG%&f*(t)G-%&COLORG6& %$RGBG$\"\"*Fcx$\"\"&Fcx$\"\"#Fcx-%*THICKNESSG6#Fby-%*AXESTICKSG6$%(DE FAULTG\"\"$-%%VIEWG6$;F(Fax;F($\"#7Fcx" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 25 "The Laplace transform of " }{XPPEDIT 18 0 "`f*`(t)" "6# -%#f*G6#%\"tG" }{TEXT -1 46 " can be obtained by making use of the res ult: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[u[a](t)*t ] = exp(-a*s)*L*[t+a];" "6#/*&%\"LG\"\"\"7#*&-&%\"uG6#%\"aG6#%\"tGF&F/ F&F&*(-%$expG6#,$*&F-F&%\"sGF&!\"\"F&F%F&7#,&F/F&F-F&F&" }{TEXT -1 2 " , " }}{PARA 0 "" 0 "" {TEXT -1 44 "which follows from the second shift formula." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "We have " }}{PARA 256 "" 0 "" {TEXT 287 1 " " }{XPPEDIT 18 0 "L*[` f*`(t)] = L*[t-u[1](t)*t];" "6#/*&%\"LG\"\"\"7#-%#f*G6#%\"tGF&*&F%F&7# ,&F+F&*&-&%\"uG6#F&6#F+F&F+F&!\"\"F&" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(s^2)-exp(-s)*L*[t+1];" "6# /%!G,&*&\"\"\"F'*$%\"sG\"\"#!\"\"F'*(-%$expG6#,$F)F+F'%\"LGF'7#,&%\"tG F'F'F'F'F+" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = 1/(s^2)-exp(-s)*(1/(s^2)+1/s);" "6#/%!G,&*&\"\"\"F '*$%\"sG\"\"#!\"\"F'*&-%$expG6#,$F)F+F',&*&F'F'*$F)F*F+F'*&F'F'F)F+F'F 'F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 42 " has period 1, it follows \+ using (i) that: " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[f(t)];" "6#*&%\"LG\"\"\"7#-%\"fG6# %\"tGF%" }{TEXT -1 4 " = " }{XPPEDIT 18 0 "1/(1-exp(-s));" "6#*&\"\" \"F$,&F$F$-%$expG6#,$%\"sG!\"\"F+F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "` `(1/(s^2)-exp(-s)*(1/(s^2)+1/s));" "6#-%!G6#,&*&\"\"\"F(*$%\"sG\"\"#! \"\"F(*&-%$expG6#,$F*F,F(,&*&F(F(*$F*F+F,F(*&F(F(F*F,F(F(F," }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = (1- exp(-s)-s*exp(-s))/(s^2*(1-exp(-s)));" "6#/%!G*&,(\"\"\"F'-%$expG6#,$% \"sG!\"\"F-*&F,F'-F)6#,$F,F-F'F-F'*&F,\"\"#,&F'F'-F)6#,$F,F-F-F'F-" } {TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "The Laplace transform of " }{XPPEDIT 18 0 "`f*`(t)" "6#-% #f*G6#%\"tG" }{TEXT -1 25 " can be determined using " }{TEXT 0 7 "lapl ace" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 199 "`f*` := t -> t-'u[1]'(t)*t:\n'`f*`(t)'=` f*`(t);\nconvert(rhs(%),Heaviside):\n`f*` := unapply(%,t):\n'`f*`(t)'= `f*`(t);\n`Laplace transform`=inttrans[laplace](`f*`(t),t,s);\n``=norm al(rhs(%));\nFs := rhs(%):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f*G6 #%\"tG,&F'\"\"\"*&-&%\"uG6#F)F&F)F'F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f*G6#%\"tG,&F'\"\"\"*&-%*HeavisideG6#,&F)!\"\"F'F)F )F'F)F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG,(*&\" \"\"F'*$)%\"sG\"\"#F'!\"\"F'*&-%$expG6#,$F*F,F'F*F,F,*&F.F'F*!\"#F," } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&,(\"\"\"!\"\"*&-%$expG6#,$%\" sGF)F(F/F(F(F+F(F(F/!\"#F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Since the periodic function " }{XPPEDIT 18 0 "f (t)" "6#-%\"fG6#%\"tG" }{TEXT -1 40 " has period 1, the Laplace transf orm of " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 42 " is obt ained by dividing the transform of " }{XPPEDIT 18 0 "`f*`(t)" "6#-%#f* G6#%\"tG" }{TEXT -1 3 " by" }{XPPEDIT 18 0 "``(1-exp(-s));" "6#-%!G6#, &\"\"\"F'-%$expG6#,$%\"sG!\"\"F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Fs/(1-exp(-s ));\n``=simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(,(\"\"\"! \"\"*&-%$expG6#,$%\"sGF'F&F-F&F&F)F&F&F-!\"#,&F&F&F)F'F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*(,(\"\"\"!\"\"*&-%$expG6#,$%\"sGF(F'F.F 'F'F*F'F'F.!\"#,&F'F(F*F'F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 " Alternatively, we can start with the piecewise definition of " } {XPPEDIT 18 0 "`f*`(t)" "6#-%#f*G6#%\"tG" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f*`(t) = PIECEWISE([t, 0 < = t and t < 1],[0, 1 <= t]);" "6#/-%#f*G6#%\"tG-%*PIECEWISEG6$7$F'31\" \"!F'2F'\"\"\"7$F.1F0F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 220 "`f*` := t -> piecewi se(t<1,t,t>=1,0):\n'`f*`(t)'=`f*`(t);\nsimplify(convert(`f*`(t),Heavis ide)):\n`f*` := unapply(%,t):\n'`f*`(t)'=`f*`(t);\n`Laplace transform` =inttrans[laplace](`f*`(t),t,s);\n``=normal(rhs(%));\nFs := rhs(%):" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f*G6#%\"tG-%*PIECEWISEG6$7$F'2F' \"\"\"7$\"\"!1F-F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f*G6#%\"tG,$ *&F'\"\"\",&F*!\"\"-%*HeavisideG6#,&F*F,F'F*F*F*F," }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%2Laplace~transformG,(*&\"\"\"F'*$)%\"sG\"\"#F'!\"\" F'*&-%$expG6#,$F*F,F'F*F,F,*&F.F'F*!\"#F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&,(\"\"\"!\"\"*&-%$expG6#,$%\"sGF)F(F/F(F(F+F(F( F/!\"#F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 " As before the Laplace transform of " }{XPPEDIT 18 0 "f(t)" "6#-% \"fG6#%\"tG" }{TEXT -1 45 " is obtained by dividing the transform of f *(" }{TEXT 299 1 "t" }{TEXT -1 4 ") by" }{XPPEDIT 18 0 "``(1-exp(-s)); " "6#-%!G6#,&\"\"\"F'-%$expG6#,$%\"sG!\"\"F-" }{TEXT -1 3 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Fs/(1-exp(-s));\n``=simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#,$*(,(\"\"\"!\"\"*&-%$expG6#,$%\"sGF'F&F-F&F&F)F&F&F-!\"#,&F&F&F)F'F 'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*(,(\"\"\"!\"\"*&-%$expG6#, $%\"sGF(F'F.F'F'F*F'F'F.!\"#,&F'F(F*F'F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "We can also obtain this result by using the integral form ula:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[f(t)];" "6#*&%\"LG\"\"\"7#-%\"fG6#%\"tGF%" } {TEXT -1 3 " = " }{XPPEDIT 18 0 "1/(1-exp(-p*s));" "6#*&\"\"\"F$,&F$F$ -%$expG6#,$*&%\"pGF$%\"sGF$!\"\"F-F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 " Int(f(t)*exp(-t*s),t = 0 .. p);" "6#-%$IntG6$*&-%\"fG6#%\"tG\"\"\"-%$e xpG6#,$*&F*F+%\"sGF+!\"\"F+/F*;\"\"!%\"pG" }{TEXT -1 3 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "f(t) = t-floor(t);" "6#/ -%\"fG6#%\"tG,&F'\"\"\"-%&floorG6#F'!\"\"" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "p = 1" "6#/%\"pG\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "f := t - > (t-floor(t)):\n'f(t)'=f(t);\nInt('f(t)'*exp(-s*t),t=0..1)/(1-exp(-s) );\n``=value(%);\n``=simplify(map(expand,rhs(%)));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"fG6#%\"tG,&F'\"\"\"-%&floorGF&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$IntG6$*&-%\"fG6#%\"tG\"\"\"-%$expG6#,$*&% \"sGF,F+F,!\"\"F,/F+;\"\"!F,*$,&F,F,-F.6#,$F2F3F3F3" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%!G,$**,(-%$expG6#%\"sG!\"\"F+\"\"\"F-F-F-F+!\"#-F)6 #,$F+F,F-,&F-F-F/F,F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*(,(- %$expG6#%\"sG!\"\"F+\"\"\"F-F-F-F+!\"#,&F(F-F-F,F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "Example 3: Laplace transform of a triangular wave" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(t) = PIECEWISE([t, 0 <= t and t < 1] ,[2-t, 1 <= t and t < 2]);" "6#/-%\"fG6#%\"tG-%*PIECEWISEG6$7$F'31\"\" !F'2F'\"\"\"7$,&\"\"#F0F'!\"\"31F0F'2F'F3" }{TEXT -1 3 " , " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 " and " } {XPPEDIT 18 0 "f(t);" "6#-%\"fG6#%\"tG" }{TEXT -1 28 " is periodic wit h period 2. 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7l=@]i/*G1s<+%Feo$\"3)3J\\>*G1sSFeo $\"3Q4FwX-b:>F-7$$\"3!>rid+pp/%Feo$\"3=!>rid+pp%F-7$$\"3q'zn!4DyuSFeo$ \"3)4nzn!4DyuF-7$$\"3SQ.sv!o\"HTFeo$\"3#RQ.sv!o\"H\"F37$$\"3'4)GPUOb$= %Feo$\"3l4)GPUOb$=F37$$\"3F,hY!H!\\SUFeo$\"3k75m/H!\\S#F37$$\"3c@$f&Qp U(H%Feo$\"3k:Kf&QpU(HF37$$\"3EPUv(pjAN%Feo$\"3rsBaxpjANF37$$\"3)H:\\pX +rS%Feo$\"3yH:\\pX+rSF37$$\"3mwXZj*yJY%Feo$\"3bmduM'*yJYF37$$\"3L+++qu D>XFeo$\"3I.+++Zd#>&F37$$\"3]mz<=;([d%Feo$\"3/l'z<=;([dF37$$\"3oKfNmd[ IYFeo$\"3vE$fNmd[I'F37$$\"3C#\\pER*e\"o%Feo$\"3cA\\pER*e\"oF37$$\"3&40 $)*=IpKZFeo$\"3[40$)*=IpK(F37$$\"3'4mz*GN(\\'*\\Feo$\"3'H[>*GN(\\'**F37$$\"3+&*fhk#))***\\ Feo$\"3#*\\*fhk#))****F37$$\"3pT+Mw\"zM+&Feo$\"35$e*fO#3_'**F37$$\"3R) 3k!)3qp+&Feo$\"39;\"f$>\"*HI**F37$$\"33N\")y**4Y5]Feo$\"3;\\'=@+!R&*)* F37$$\"3m30e$H)QQ]Feo$\"3T8\\>kq6;'*F37$$\"3M\")GP(e:j1&Feo$\"3a'=ri7W oL*F37$$\"3wridabx>^Feo$\"3T#GPUXWA!))F37$$\"3Gh'z<_NK<&Feo$\"3;(Q.AyW wE)F37$$\"3fFw&\\%ffK_Feo$\"37CPU]0/uwF37$$\"3)QfN\"oj&>H&Feo$\"33hSk= jV!3(F37$$\"3=HP#H8eLM&Feo$\"393Fwq'=kc'F37$$\"3ej=r(*)fZR&Feo$\"31k8) G-,C0'F37$$\"3#[Z30O,JX&Feo$\"3u^_\"\\R')*oaF37$$\"3<&30L#GW6bFeo$\"3I [\"\\pwrb)[F37$$\"3_?K4LZhkbFeo$\"3y%zn!pE&QN%F37$$\"3)eN\")Gk'yfgFeo$\"3-;YZ3X!*>fF-7$$\"3DKf&QMqV6'Feo$\"3bA$f&QMqV6F37$$\"3 M=riU;aphFeo$\"3]$=riU;ap\"F37$$\"34oS9w*4XA'Feo$\"3!4oS9w*4XAF37$$\"3 %p,h'4$y%ziFeo$\"3Up,h'4$y%z#F37$$\"3C!R.(pcGKjFeo$\"3T-R.(pcGK$F37$$ \"3kiduHI4&Q'Feo$\"3]EwX(HI4&QF37$$\"3K\")G([^YDW'Feo$\"3D8)G([^YDWF37 $$\"3++++++++lFeo$\"3++++++++]F3-%&COLORG6&%$RGBG$\"\"*!\"\"$\"\"%F\\f mF(-%*THICKNESSG6#\"\"#-%+AXESLABELSG6$%\"tG%%f(t)G-%*AXESTICKSG6$%(DE FAULTG\"\"$-%%VIEWG6$;F($\"#lF\\fm;$!\"#F\\fm$\"#7F\\fm" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 60 "The associated (non-periodic) function wh ich coincides with " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 52 " over the \"first period\" and is zero thereafter is: " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`f*`(t) = PIECEWISE([t, 0 <= t and t < 1],[2-t, 1 <= t \+ and t < 2],[0, 2 <= t]);" "6#/-%#f*G6#%\"tG-%*PIECEWISEG6%7$F'31\"\"!F '2F'\"\"\"7$,&\"\"#F0F'!\"\"31F0F'2F'F37$F.1F3F'" }{TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "`` = t+u[1](t)*(2-2*t)+u[2](t)*(t-2);" "6#/%!G,(%\"tG\" \"\"*&-&%\"uG6#F'6#F&F',&\"\"#F'*&F/F'F&F'!\"\"F'F'*&-&F+6#F/6#F&F',&F &F'F/F1F'F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 1 "u" }{TEXT -1 4 " .. " }{HYPERLNK 17 "u" 1 "" "u" }{TEXT -1 88 " from the subsection which follows the examples, and which implements the step function " } {XPPEDIT 18 0 "u[a](t)=`` " "6#/-&%\"uG6#%\"aG6#%\"tG%!G" }{TEXT 297 7 "u[a](t)" }{TEXT -1 42 " may be used to set up the Maple function " }{TEXT 0 4 "`f*`" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "`f*`(t)" "6#-%#f *G6#%\"tG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 204 "`f*` := t -> t+'u[1]'(t)*(2-2*t)+' u[2]'(t)*(t-2):\n'`f*`(t)'=`f*`(t);\n``=convert(rhs(%),piecewise);\npl ot(`f*`(t),t=0..5.5,0..1.2,thickness=2,color=COLOR(RGB,.9,.1,0),\n \+ labels=[t,`f*(t)`],ytickmarks=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%#f*G6#%\"tG,(F'\"\"\"*&-&%\"uG6#F)F&F),&\"\"#F)*&F0F)F'F)!\"\"F)F)* &-&F-6#F0F&F),&F0F2F'F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%* PIECEWISEG6%7$%\"tG2F)\"\"\"7$,&\"\"#F+F)!\"\"2F)F.7$\"\"!1F.F)" }} {PARA 13 "" 1 "" {GLPLOT2D 593 155 155 {PLOTDATA 2 "6(-%'CURVESG6#7fo7 $$\"\"!F)F(7$$\"+&R\"!#5F/7$$\"+IrR?C#F1F67$$\"+2C\\GGF1F97$$\"+MS.:MF1F<7$$\"+SOY0SF1F?7$$\"+ZK*ef%F1FB 7$$\"+Xm^$=&F1FE7$$\"+T+9rdF1FH7$$\"+]@%fJ'F1FK7$$\"+iUugoF1FN7$$\"+!H b[U(F1FQ7$$\"+?j'*))zF1FT7$$\"+b2Ps&)F1FW7$$\"+'=vd:*F1FZ7$$\"+5>aY%*F 1Fgn7$$\"+N'3tt*F1Fjn7$$\"++?p#))*F1F]o7$$\"+Ov!G+\"!\"*$\"*kC>(**Fbo7 $$\"+seM<5Fbo$\"*GTl#)*Fbo7$$\"+3U)=.\"Fbo$\"*#z:\"o*Fbo7$$\"+r?q\"4\" Fbo$\"*HzH3*Fbo7$$\"+M*>::\"Fbo$\"*m+[[)Fbo7$$\"+Sv?/7Fbo$\"*gCz&zFbo7 $$\"+Z^*oD\"Fbo$\"*`[5V(Fbo7$$\"+7*3iJ\"Fbo$\"*)3\"z$oFbo7$$\"+yE_v8Fb o$\"*AtZC'Fbo7$$\"+#**z]V\"Fbo$\"*3+#\\cFbo7$$\"+1tj%\\\"Fbo$\"*%pi`]F bo7$$\"+9=._:Fbo$\"*'=ozWFbo7$$\"+@jU4;Fbo$\"*zOd!RFbo7$$\"+Kgah;Fbo$ \"*oRXQ$Fbo7$$\"+Vdm8E%>Fbo$\") _4QdFbo7$$\"+!yaJ(>Fbo$\")?_%o#Fbo7$$\"+60p.?FboF(7$$\"+UiAM?FboF(7$$ \"+u>wk?FboF(7$$\"+n0#)=@FboF(7$$\"+g\"zG<#FboF(7$$\"+\\3]\"H#FboF(7$$ \"+[nX/CFboF(7$$\"+zVJADFboF(7$$\"+wWaIEFboF(7$$\"+c_GZFFboF(7$$\"+=ha oGFboF(7$$\"+aS5uHFboF(7$$\"+=&4\")3$FboF(7$$\"+<$))e?$FboF(7$$\"+$*=6 @LFboF(7$$\"+HmfKMFboF(7$$\"+?9QcNFboF(7$$\"+gvgnOFboF(7$$\"+0_O'y$Fbo F(7$$\"+$GxR*QFboF(7$$\"+3di6SFboF(7$$\"+2PKATFboF(7$$\"+7].QUFboF(7$$ \"+jR;^VFboF(7$$\"+\"\\*fpWFboF(7$$\"+!)pm$e%FboF(7$$\"+juJ+ZFboF(7$$ \"+A?+;[FboF(7$$\"+;TIA\\FboF(7$$\"+7$QT/&Fbo$\"\"%Fbo7$$\"+q#3J:&FboF (7$$\"+tHHp_Fbo$!\"%Fbo7$$\"+fl\\!Q&Fbo$!\"#Fbo7$$\"#b!\"\"F(-%+AXESLA BELSG6$%\"tG%&f*(t)G-%&COLORG6&%$RGBG$\"\"*F`\\l$\"\"\"F`\\lF(-%*THICK NESSG6#\"\"#-%*AXESTICKSG6$%(DEFAULTG\"\"$-%%VIEWG6$;F(F^\\l;F($\"#7F` \\l" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1 " }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "The Laplace transform of \+ " }{XPPEDIT 18 0 "`f*`(t)" "6#-%#f*G6#%\"tG" }{TEXT -1 60 " can be obt ained by making use of the second shift formula: " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[u[a](t)*g(t)] = exp(-a*s)*L*[g(t+a )];" "6#/*&%\"LG\"\"\"7#*&-&%\"uG6#%\"aG6#%\"tGF&-%\"gG6#F/F&F&*(-%$ex pG6#,$*&F-F&%\"sGF&!\"\"F&F%F&7#-F16#,&F/F&F-F&F&" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "We have \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[`f*`(t)] = L*[ t+u[1](t)*(2-2*t)+u[2](t)*(t-2)];" "6#/*&%\"LG\"\"\"7#-%#f*G6#%\"tGF&* &F%F&7#,(F+F&*&-&%\"uG6#F&6#F+F&,&\"\"#F&*&F6F&F+F&!\"\"F&F&*&-&F26#F6 6#F+F&,&F+F&F6F8F&F&F&" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(s^2)+exp(-s)*L*[-2*t]+exp(-2*s)*L*[t];" "6#/%!G,(*&\"\"\"F'*$%\"sG\"\"#!\"\"F'*(-%$expG6#,$F)F+F'%\"LGF'7#,$*& F*F'%\"tGF'F+F'F'*(-F.6#,$*&F*F'F)F'F+F'F1F'7#F5F'F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(s^2)-2*exp( -s)/(s^2)+exp(-2*s)/(s^2);" "6#/%!G,(*&\"\"\"F'*$%\"sG\"\"#!\"\"F'*(F* F'-%$expG6#,$F)F+F'*$F)F*F+F+*&-F.6#,$*&F*F'F)F'F+F'*$F)F*F+F'" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` \+ = (1-2*exp(-s)+exp(-2*s))/(s^2);" "6#/%!G*&,(\"\"\"F'*&\"\"#F'-%$expG6 #,$%\"sG!\"\"F'F/-F+6#,$*&F)F'F.F'F/F'F'*$F.F)F/" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = (1-exp(-s))^2/( s^2);" "6#/%!G*&,&\"\"\"F'-%$expG6#,$%\"sG!\"\"F-\"\"#*$F,F.F-" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "f (t)" "6#-%\"fG6#%\"tG" }{TEXT -1 32 " has period 2, it follows that \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[f(t)];" "6#*&% \"LG\"\"\"7#-%\"fG6#%\"tGF%" }{TEXT -1 4 " = " }{XPPEDIT 18 0 "1/(1-e xp(-2*s));" "6#*&\"\"\"F$,&F$F$-%$expG6#,$*&\"\"#F$%\"sGF$!\"\"F-F-" } {TEXT -1 1 " " }{XPPEDIT 18 0 "``((1-exp(-s))^2/(s^2));" "6#-%!G6#*&,& \"\"\"F(-%$expG6#,$%\"sG!\"\"F.\"\"#*$F-F/F." }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (1-exp(-s))^2/(s^2*(1+ exp(-s))*(1-exp(-s)));" "6#/%!G*&,&\"\"\"F'-%$expG6#,$%\"sG!\"\"F-\"\" #*(F,F.,&F'F'-F)6#,$F,F-F'F',&F'F'-F)6#,$F,F-F-F'F-" }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = (1-exp(-s))/(s^2*(1+exp(-s)));" "6#/%!G*&,&\"\"\"F '-%$expG6#,$%\"sG!\"\"F-F'*&F,\"\"#,&F'F'-F)6#,$F,F-F'F'F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 " The Laplace transform of " }{XPPEDIT 18 0 "`f*`(t)" "6#-%#f*G6#%\"tG" }{TEXT -1 25 " can be determined using " }{TEXT 0 7 "laplace" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 221 "`f*` := t -> t+'u[1]'(t)*(2-2*t)+'u[2]'(t)*(t-2):\n' `f*`(t)'=`f*`(t);\nconvert(rhs(%),Heaviside):\n`f*` := unapply(%,t):\n '`f*`(t)'=`f*`(t);\n`Laplace transform`=inttrans[laplace](`f*`(t),t,s) ;\n``=normal(rhs(%));\nFs := rhs(%):" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/-%#f*G6#%\"tG,(F'\"\"\"*&-&%\"uG6#F)F&F),&\"\"#F)*&F0F)F'F)!\"\"F)F )*&-&F-6#F0F&F),&F0F2F'F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f *G6#%\"tG,,F'\"\"\"*&\"\"#F)-%*HeavisideG6#,&F)!\"\"F'F)F)F)*(F+F)F,F) F'F)F0*&F+F)-F-6#,&F+F0F'F)F)F0*&F3F)F'F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG,(*&\"\"\"F'*$)%\"sG\"\"#F'!\"\"F' *(F+F'-%$expG6#,$F*F,F'F*!\"#F,*&-F/6#,$*&F+F'F*F'F,F'F*F2F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&,(\"\"\"!\"\"*&\"\"#F(-%$expG6#,$%\" sGF)F(F(-F-6#,$*&F+F(F0F(F)F)F(F0!\"#F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Since the periodic function " } {XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 40 " has period 2, th e Laplace transform of " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" } {TEXT -1 42 " is obtained by dividing the transform of " }{XPPEDIT 18 0 "`f*`(t)" "6#-%#f*G6#%\"tG" }{TEXT -1 3 " by" }{XPPEDIT 18 0 "``(1-e xp(-2*s));" "6#-%!G6#,&\"\"\"F'-%$expG6#,$*&\"\"#F'%\"sGF'!\"\"F/" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Fs/(1-exp(-2*s));\n``=simplify(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,$*(,(\"\"\"!\"\"*&\"\"#F&-%$expG6#,$%\"sGF'F&F& -F+6#,$*&F)F&F.F&F'F'F&F.!\"#,&F&F&F/F'F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*(,&-%$expG6#,$%\"sG!\"\"\"\"\"F.F-F.,&F(F.F.F.F- F,!\"#F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Alternatively, we \+ can start with the piecewise definition of " }{XPPEDIT 18 0 "`f*`(t)" "6#-%#f*G6#%\"tG" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f*`(t) = PIECEWISE([t, 0 <= t and t < 1],[2-t, 1 <= t and t < 2],[0, 2 <= t]);" "6#/-%#f*G6#%\"tG-%*PIECEWISEG6%7$F'31\"\"! F'2F'\"\"\"7$,&\"\"#F0F'!\"\"31F0F'2F'F37$F.1F3F'" }{TEXT -1 3 " . " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 228 "`f*` := t -> piecewise(t<1,t,t<2,2-t,t>=2,0):\n'`f*`(t)'=`f*`(t );\nsimplify(convert(`f*`(t),Heaviside)):\n`f*` := unapply(%,t):\n'`f* `(t)'=`f*`(t);\n`Laplace transform`=inttrans[laplace](`f*`(t),t,s);\n` `=normal(rhs(%));\nFs := rhs(%):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %#f*G6#%\"tG-%*PIECEWISEG6%7$F'2F'\"\"\"7$,&\"\"#F-F'!\"\"2F'F07$\"\"! 1F0F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f*G6#%\"tG,,F'\"\"\"*&\" \"#F)-%*HeavisideG6#,&F)!\"\"F'F)F)F)*(F+F)F,F)F'F)F0*&F+F)-F-6#,&F+F0 F'F)F)F0*&F3F)F'F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~tra nsformG,(*&\"\"\"F'*$)%\"sG\"\"#F'!\"\"F'*(F+F'-%$expG6#,$F*F,F'F*!\"# F,*&-F/6#,$*&F+F'F*F'F,F'F*F2F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%! G,$*&,(\"\"\"!\"\"*&\"\"#F(-%$expG6#,$%\"sGF)F(F(-F-6#,$*&F+F(F0F(F)F) F(F0!\"#F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 " As before the Laplace transform of " }{XPPEDIT 18 0 "f(t)" "6# -%\"fG6#%\"tG" }{TEXT -1 42 " is obtained by dividing the transform of " }{XPPEDIT 18 0 "`f*`(t)" "6#-%#f*G6#%\"tG" }{TEXT -1 3 " by" } {XPPEDIT 18 0 "``(1-exp(-2*s));" "6#-%!G6#,&\"\"\"F'-%$expG6#,$*&\"\"# F'%\"sGF'!\"\"F/" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Fs/(1-exp(-2*s));\n``=simpli fy(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(,(\"\"\"!\"\"*&\"\"#F&-% $expG6#,$%\"sGF'F&F&-F+6#,$*&F)F&F.F&F'F'F&F.!\"#,&F&F&F/F'F'F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*(,&-%$expG6#,$%\"sG!\"\"\"\"\"F .F-F.,&F(F.F.F.F-F,!\"#F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "We can also obtain this result by using the integral formula:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[f(t)];" "6#*&%\"LG\"\"\" 7#-%\"fG6#%\"tGF%" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "1/(1-exp(-p*s)); " "6#*&\"\"\"F$,&F$F$-%$expG6#,$*&%\"pGF$%\"sGF$!\"\"F-F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(t)*exp(-t*s),t = 0 .. p);" "6#-%$IntG6$*&- %\"fG6#%\"tG\"\"\"-%$expG6#,$*&F*F+%\"sGF+!\"\"F+/F*;\"\"!%\"pG" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 " p=2" "6#/%\"pG\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "T he function " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 16 " c oincides with " }{XPPEDIT 18 0 "`f*`(t)" "6#-%#f*G6#%\"tG" }{TEXT -1 19 " over the interval " }{XPPEDIT 18 0 "[0,2]" "6#7$\"\"!\"\"#" } {TEXT -1 5 ", so " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 37 " in the integrand can be replaced by " }{XPPEDIT 18 0 "`f*`(t)" "6 #-%#f*G6#%\"tG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f*`(t) = t+u[1](t)*( 2-2*t)+u[2](t)*(t-2);" "6#/-%#f*G6#%\"tG,(F'\"\"\"*&-&%\"uG6#F)6#F'F), &\"\"#F)*&F1F)F'F)!\"\"F)F)*&-&F-6#F16#F'F),&F'F)F1F3F)F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = PIECEWISE([t, 0 <= t and t < 1],[2-t, 1 <= t a nd t < 2],[0, 2 <= t]);" "6#/%!G-%*PIECEWISEG6%7$%\"tG31\"\"!F)2F)\"\" \"7$,&\"\"#F.F)!\"\"31F.F)2F)F17$F,1F1F)" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "`f*` := t -> piecewise(t<1,t,t<2,2-t,t>=2,0):\n'`f*`(t)'=`f*`(t);\nInt('`f *`(t)'*exp(-s*t),t=0..2)/(1-exp(-2*s));\n``=value(%);\n``=simplify(rhs (%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f*G6#%\"tG-%*PIECEWISEG6% 7$F'2F'\"\"\"7$,&\"\"#F-F'!\"\"2F'F07$\"\"!1F0F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$IntG6$*&-%#f*G6#%\"tG\"\"\"-%$expG6#,$*&%\"sGF,F+F ,!\"\"F,/F+;\"\"!\"\"#*$,&F,F,-F.6#,$*&F7F,F2F,F3F3F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*(,(\"\"\"!\"\"*&\"\"#F(-%$expG6#,$%\"sGF)F( F(-F-6#,$*&F+F(F0F(F)F)F(F0!\"#,&F(F(F1F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*(,&-%$expG6#,$%\"sG!\"\"\"\"\"F.F-F.,&F(F.F.F.F- F,!\"#F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 74 "Example 4: Laplace transform of the full-wave rectification of a sine wave" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "f(t) = abs(sin(t));" "6#/-%\"fG6#%\"tG-%$absG6#-%$sinG6 #F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "plot(abs(sin(t)),t=0..13.5,-0.2..1.2,thick ness=2,numpoints=100,\n color=COLOR(RGB,0,.6,.8),labels=[t,`f(t) `],ytickmarks=3);" }}{PARA 13 "" 1 "" {GLPLOT2D 547 150 150 {PLOTDATA 2 "6(-%'CURVESG6#7_v7$$\"\"!F)F(7$$\"3g!44449O8(!#>$\"3z4ux?`cFrF-7$$ \"37====GsE9!#=$\"3HZ#HX\\()=U\"F37$$\"3y!44fw9u/#F3$\"3C=`@H/9L?F37$$ \"3Ujjj8n5oEF3$\"3)=&f5KIcOEF37$$\"3q*****\\sRhO$F3$\"3D)4TF$*HHI$F37$ $\"3'fjjjtsT1%F3$\"3'p?c!Q)4K&RF37$$\"3KjjjjF$ow%F3$\"3yD2vRsM)e%F37$$ \"3o!444z#\\paF3$\"3XmX$G')R3?&F37$$\"3+sss(>8)ohF3$\"3ygh\"Q7T\\y&F37 $$\"3I`aa/O8ooF3$\"32(40Uri2M'F37$$\"3g*3449$\\;vF3$\"3c!Q&GrhWGoF37$$ \"3!fsssn_[;)F3$\"3owTEP\\V(G(F37$$\"3&p\"==o,`2&*F3$\"3I/&*z7H`Q\")F3 7$$\"3-+++:Hh*3\"!#<$\"3%zR>)o#yW'))F37$$\"3bsss([F!G7Fjo$\"3()H9@3oF= %*F37$$\"3;444*)QSq8Fjo$\"3u[[Qq`))*z*F37$$\"34+++&\\1JV\"Fjo$\"3B+@j; sN0**F37$$\"3!344454e\\\"Fjo$\"3'*\\[\">v(*=(**F37$$\"3'*****\\UK5J:Fj o$\"3MH`aHL7#***F37$$\"37444%Q(Rm:Fjo$\"3Yb9nYK!*****F37$$\"3F==oD:p,; Fjo$\"37.)RI\"yA&***F37$$\"3@FFFnc)pj\"Fjo$\"3s,L7_G5y**F37$$\"3isss#z jyq\"Fjo$\"39YveF&4i!**F37$$\"3/====>uyFjo$\"3VUE=F>E7%*F37$$\"3ZXXXgQSR?Fjo$\"3z(G8a_!)>#*)F37$$\"3mjjj. b\"p=#Fjo$\"3C)y9*o@Fh\")F37$$\"3'*344fjR\\AFjo$\"3Cc^.kC_%y(F37$$\"3D aaa9s(=J#Fjo$\"3?JMWzHRxtF37$$\"3/XX&HIdXQ#Fjo$\"3AIDEh`pnoF37$$\"3GOO O\"RPsX#Fjo$\"3m,jTGet@jF37$$\"3\"pssZer:_#Fjo$\"35@mkyI_5eF37$$\"3+== =yd!fe#Fjo$\"33\"*pj:&p_F&F37$$\"3#pssZe!\\cEFjo$\"3'>qbua')Hm%F37$$\" 3IOOO\"Rvqs#Fjo$\"3Mme3A7[FSF37$$\"3w344>))G%z#Fjo$\"3A1o3*fQOS$F37$$ \"3m\"==oC-:'GFjo$\"3CB8aL`UkFF37$$\"3)*****\\2CjJHFjo$\"3%>C#G\">5U3# F37$$\"3I===oDw,IFjo$\"3i/m$[g[PR\"F37$$\"3]XX&z%R;mIFjo$\"34F3 7$$\"3*fjj8&p!QT$Fjo$\"3C\\2)yc[')o#F37$$\"35XXX&G=mZ$Fjo$\"3%yFKwwLzG $F37$$\"3@aaa>'H%RNFjo$\"3eIl'eY^U(QF37$$\"3Djj8\"ensg$Fjo$\"37SzhS'f- \\%F37$$\"3usssUb5vOFjo$\"3!*4l?8;h&3&F37$$\"3!)*****\\z)=XPFjo$\"3G![ )R&Rogn&F37$$\"3%osss/s_\"QFjo$\"3g'G_eld'QiF37$$\"3?\"==o$[$Q)QFjo$\" 3em(e5(*>%fnF37$$\"3+OOOEwR_RFjo$\"3'=BT6n>%[sF37$$\"3E<=o]ct=SFjo$\"3 %QF49$R:*o(F37$$\"3S*****\\nt]3%Fjo$\"3'*Hj4#oig4)F37$$\"3aNOO1yQKUFjo $\"3?%)>S1Q%*p))F37$$\"3]WXX0hvkVFjo$\"3v!QBIi<=S*F37$$\"3I====v31XFjo $\"3I9Th/A&zy*F37$$\"3Ijj8^67qXFjo$\"3'QbK;!*p*)*)*F37$$\"3J344%yaTj%F jo$\"3GuNZeETp**F37$$\"3A***\\7id\"pYFjo$\"3S*o_Kqc1***F37$$\"39!44%e/ ;/ZFjo$\"3]o1/f9m****F37$$\"31\"=obHj\"RZFjo$\"3_H'\\V*eT'***F37$$\"3' GFFF8mTx%Fjo$\"33M\\B&)R#4)**F37$$\"3l34f;g.S[Fjo$\"3!pRy,tT'=**F37$$ \"3WWXX+f!f!\\Fjo$\"3vxf$yhRL\")*F37$$\"3m\"==o\">hV]Fjo$\"3!prQ;_`kX* F37$$\"3U<===ZCy^Fjo$\"3?xCYg]PM*)F37$$\"3z344%y#>>`Fjo$\"3W$Hs&\\#[Z@ )F37$$\"3=OO'QlnqQ&Fjo$\"3e#4b!pq24yF37$$\"3cjjjBD%\\X&Fjo$\"3K<(GjHVu O(F37$$\"3WOOO\"QaV_&Fjo$\"3/!o%\\uet!)oF37$$\"3U344Riw$f&Fjo$\"3/gp:M .*3O'F37$$\"3M34fJLgicFjo$\"3YoZ2&>(3:eF37$$\"3:444C/WJdFjo$\"3Ej.1\") )R0&31ZX$F37$$\"3Taaak9%H+'Fjo$\"3 k*oaj')**ew#F37$$\"3sEFF#4$yngFjo$\"30?D-bOSP@F37$$\"3#*******>ZiKhFjo $\"39%f\"F37$$\"3Yjj8m'4nZ'Fjo$\"3r6R;.l=B>F37$$\"3EOOOcP =]lFjo$\"3!e8.=Ku$QEF37$$\"3sEFx>p!Qh'Fjo$\"3!R2feP5jC$F37$$\"3=<==$3I un'Fjo$\"3/n,z*456%QF37$$\"3%>FFF%Hk\\nFjo$\"3EMR*o!>D(\\%F37$$\"3qEFF -e&=#oFjo$\"30B\\*p4_*H^F37$$\"3W\"==V)*pw)oFjo$\"3IrjBm9Q$o&F37$$\"3? 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" }}{PARA 0 "" 0 "" {TEXT -1 60 "The associated (non-per iodic) function which coincides with " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG 6#%\"tG" }{TEXT -1 38 " over the \"first period\", that is for " } {XPPEDIT 18 0 "0<=t" "6#1\"\"!%\"tG" }{XPPEDIT 18 0 "`` " 0 "" {MPLTEXT 1 0 207 "`f*` := t -> sin(t)-'u[Pi]'(t)*sin(t):\n'`f*`(t)'=`f*`(t);\n``=co nvert(rhs(%),piecewise);\nplot(`f*`(t),t=0..8.5,0..1.2,thickness=2,col or=COLOR(RGB,.6,.1,.8),\n labels=[t,`f*(t)`],ytickmarks=3,numpoint s=75);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f*G6#%\"tG,&-%$sinGF&\" \"\"*&-&%\"uG6#%#PiGF&F+F)F+!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %!G-%*PIECEWISEG6$7$-%$sinG6#%\"tG2F,%#PiG7$\"\"!1F.F," }}{PARA 13 "" 1 "" {GLPLOT2D 539 148 148 {PLOTDATA 2 "6(-%'CURVESG6#7\\q7$$\"\"!F)F( 7$$\"+0_%*3g!#6$\"+K(H`+'F-7$$\"+T!*y,7!#5$\"+H#)*))>\"F37$$\"+]TiC&F3$\"+H*[b'\\F37$$\"+](>`y&F3$\"+')R&zY&F37$$\"+S/YJjF3$\"+$G Ro\"fF37$$\"+J6gxoF3$\"+I33[jF37$$\"+M_f3!)F3$\"+;mazrF37$$\"+`4Fy\"*F 3$\"+MN$G%zF37$$\"+^&>W.\"!\"*$\"+u(ecf)F37$$\"+A#\\V:\"F_p$\"+W!>`9*F 37$$\"+UL)*f7F_p$\"+sC&3_*F37$$\"+UB!*y8F_p$\"+\"yZk\")*F37$$\"+))fgQ9 F_p$\"+Fdv7**F37$$\"+M'4$)\\\"F_p$\"+)*)RP(**F37$$\"+(Rxq_\"F_p$\"+)zW /***F37$$\"+g^%eb\"F_p$\"+MB)))***F37$$\"+BHh%e\"F_p$\"+Cb/****F37$$\" +'o!Q8;F_p$\"+KU$4***F37$$\"+c%Gcm\"F_p$\"+D!o]&**F37$$\"+Di(yr\"F_p$ \"+jB.#*)*F37$$\"+wQ+!y\"F_p$\"+=!e>y*F37$$\"+G:8U=F_p$\"+s*QTj*F37$$ \"+p?RZ>F_p$\"+4'>#*H*F37$$\"+,^$)p?F_p$\"+E*H/y)F37$$\"+Jz@y@F_p$\"+2 PA6#)F37$$\"+u58(H#F_p$\"+tj8wuF37$$\"+1XO5CF_p$\"+gJ%yn'F37$$\"+9<^GD F_p$\"+L-!Rv&F37$$\"+B(fFe#F_p$\"+4.(>I&F37$$\"+Kx+PEF_p$\"+E8WM[F37$$ \"+Q:_&p#F_p$\"+NVC9VF37$$\"+X`.aFF_p$\"+d,GzPF37$$\"+XZ\"[\"GF_p$\"+n 6$*4KF37$$\"+XTfvGF_p$\"+1zsGEF37$$\"+\"z-&GHF_p$\"+6)3[6#F37$$\"+P9T \")HF_p$\"+g5(\\f\"F37$$\"+CUbQIF_p$\"+8?cG5F37$$\"+7qp&4$F_p$\"+yI&ze %F-7$$\"+eS@DJF_p$\"+KjyP;F-7$$\"+.6taJF_pF(7$$\"+[\"[U=$F_pF(7$$\"+%> lP@$F_pF(7$$\"+M&=:F$F_pF(7$$\"+v=FHLF_pF(7$$\"+H0.TMF_pF(7$$\"+g%>^c$ F_pF(7$$\"+$))=mn$F_pF(7$$\"+;$ocz$F_pF(7$$\"+(zWN!RF_pF(7$$\"+%G#[@SF _pF(7$$\"+pAXKTF_pF(7$$\"+xyW[UF_pF(7$$\"+'ya=O%F_pF(7$$\"+58e![%F_pF( 7$$\"+k!H\\f%F_pF(7$$\"+eh'=r%F_pF(7$$\"+`\\$y#[F_pF(7$$\"+K#)RM\\F_pF (7$$\"+w<`c]F_pF(7$$\"+t%pd;&F_pF(7$$\"+U'RAG&F_pF(7$$\"+bkr$R&F_pF(7$ $\"+Ir\\_\\yfF_pF(7$$\"+y-9$3'F_pF(7$$\"+&y=B?'F_pF(7$$\"+R$RuJ'F_pF(7$$\"+) o%[KkF_pF(7$$\"+0o5ZlF_pF(7$$\"+wFAdmF_pF(7$$\"+d5EwnF_pF(7$$\"+rJr*)o F_pF(7$$\"+xh64qF_pF(7$$\"+_Y?v#)F_pF(7$$ \"+/Gn#Q)F_pF(7$$\"#&)!\"\"F(-%+AXESLABELSG6$%\"tG%&f*(t)G-%&COLORG6&% $RGBG$\"\"'F^cl$\"\"\"F^cl$\"\")F^cl-%*THICKNESSG6#\"\"#-%*AXESTICKSG6 $%(DEFAULTG\"\"$-%%VIEWG6$;F(F\\cl;F($\"#7F^cl" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "The Laplace transform of " }{XPPEDIT 18 0 "`f*`(t)" "6#-%#f*G6#%\"tG" }{TEXT -1 60 " can be obtained by making use of the \+ second shift formula: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[u[a](t)*g(t)] = exp(-a*s)*L*[g(t+a)];" "6#/*&%\"LG\"\"\"7#*&- &%\"uG6#%\"aG6#%\"tGF&-%\"gG6#F/F&F&*(-%$expG6#,$*&F-F&%\"sGF&!\"\"F&F %F&7#-F16#,&F/F&F-F&F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 8 "We have " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[`f*`(t)] = L*[sin(t)-u[Pi](t)*sin(t)];" "6 #/*&%\"LG\"\"\"7#-%#f*G6#%\"tGF&*&F%F&7#,&-%$sinG6#F+F&*&-&%\"uG6#%#Pi G6#F+F&-F06#F+F&!\"\"F&" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(s^2+1)-exp(-Pi*s)*L*[sin(t+Pi)];" "6#/%! G,&*&\"\"\"F',&*$%\"sG\"\"#F'F'F'!\"\"F'*(-%$expG6#,$*&%#PiGF'F*F'F,F' %\"LGF'7#-%$sinG6#,&%\"tGF'F3F'F'F," }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` \+ = 1/(s^2+1)+exp(-Pi*s)*L*[sin(t)];" "6#/%!G,&*&\"\"\"F',&*$%\"sG\"\"#F 'F'F'!\"\"F'*(-%$expG6#,$*&%#PiGF'F*F'F,F'%\"LGF'7#-%$sinG6#%\"tGF'F' " }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 " `` = 1/(s^2+1)+exp(-Pi*s)/(s^2+1);" "6#/%!G,&*&\"\"\"F',&*$%\"sG\"\"#F 'F'F'!\"\"F'*&-%$expG6#,$*&%#PiGF'F*F'F,F',&*$F*F+F'F'F'F,F'" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = (1+exp(-Pi*s))/(s^2+1);" "6#/%!G*&,&\" \"\"F'-%$expG6#,$*&%#PiGF'%\"sGF'!\"\"F'F',&*$F.\"\"#F'F'F'F/" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 17 "It follows that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[f(t)] = 1/(1-exp(-Pi*s)) ;" "6#/*&%\"LG\"\"\"7#-%\"fG6#%\"tGF&*&F&F&,&F&F&-%$expG6#,$*&%#PiGF&% \"sGF&!\"\"F5F5" }{XPPEDIT 18 0 "``((1+exp(-Pi*s))/(s^2+1));" "6#-%!G6 #*&,&\"\"\"F(-%$expG6#,$*&%#PiGF(%\"sGF(!\"\"F(F(,&*$F/\"\"#F(F(F(F0" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = (1+exp(-Pi*s))/((1-exp(-Pi*s))*(s^2+1));" "6#/%!G*&,&\"\"\"F'-%$exp G6#,$*&%#PiGF'%\"sGF'!\"\"F'F'*&,&F'F'-F)6#,$*&F-F'F.F'F/F/F',&*$F.\" \"#F'F'F'F'F/" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = (exp(Pi*s)+1)/( (exp(Pi*s)-1)*(s^2+1));" "6#/%!G*&,&-%$expG6#*&%#PiG\"\"\"%\"sGF,F,F,F ,F,*&,&-F(6#*&F+F,F-F,F,F,!\"\"F,,&*$F-\"\"#F,F,F,F,F3" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "The \+ Laplace transform of " }{XPPEDIT 18 0 "`f*`(t)" "6#-%#f*G6#%\"tG" } {TEXT -1 25 " can be determined using " }{TEXT 0 7 "laplace" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 210 "`f*` := t -> sin(t)-'u[Pi]'(t)*sin(t):\n'`f*`(t)'=`f *`(t);\nconvert(rhs(%),Heaviside):\n`f*` := unapply(%,t):\n'`f*`(t)'=` f*`(t);\n`Laplace transform`=inttrans[laplace](`f*`(t),t,s);\n``=norma l(rhs(%));\nFs := rhs(%):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f*G6# %\"tG,&-%$sinGF&\"\"\"*&-&%\"uG6#%#PiGF&F+F)F+!\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%#f*G6#%\"tG,&-%$sinGF&\"\"\"*&-%*HeavisideG6#,&%#P iG!\"\"F'F+F+F)F+F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~tran sformG,&*&\"\"\"F',&*$)%\"sG\"\"#F'F'F'F'!\"\"F'*&-%$expG6#,$*&F+F'%#P iGF'F-F'F(F-F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&,&\"\"\"F'-%$e xpG6#,$*&%\"sGF'%#PiGF'!\"\"F'F',&*$)F-\"\"#F'F'F'F'F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Since the periodic f unction " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 12 " has p eriod " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 27 ", the Laplace tran sform of " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 42 " is o btained by dividing the transform of " }{XPPEDIT 18 0 "`f*`(t)" "6#-%# f*G6#%\"tG" }{TEXT -1 3 " by" }{XPPEDIT 18 0 "``(1-exp(-Pi*s));" "6#-% !G6#,&\"\"\"F'-%$expG6#,$*&%#PiGF'%\"sGF'!\"\"F/" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Fs/(1-exp(-Pi*s));\n``=simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(,&\"\"\"F%-%$expG6#,$*&%\"sGF%%#PiGF%!\"\"F%F%,&*$)F+\"\"#F%F% F%F%F-,&F%F%F&F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*(,&\"\"\" F(-%$expG6#,$*&%\"sGF(%#PiGF(!\"\"F(F(,&*$)F.\"\"#F(F(F(F(F0,&F(F0F)F( F0F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Alternatively, we can s tart with the piecewise definition of " }{XPPEDIT 18 0 "`f*`(t)" "6#-% #f*G6#%\"tG" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`f*`(t) = PIECEWISE([sin*t, 0 <= t and t < Pi],[0, Pi < = t]);" "6#/-%#f*G6#%\"tG-%*PIECEWISEG6$7$*&%$sinG\"\"\"F'F.31\"\"!F'2 F'%#PiG7$F11F3F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 227 "`f*` := t -> piecewise(t=Pi,0):\n'`f*`(t)'=`f*`(t);\nsimplify(convert(`f*`(t),Heavis ide)):\n`f*` := unapply(%,t):\n'`f*`(t)'=`f*`(t);\n`Laplace transform` =inttrans[laplace](`f*`(t),t,s);\n``=normal(rhs(%));\nFs := rhs(%):" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f*G6#%\"tG-%*PIECEWISEG6$7$-%$sin GF&2F'%#PiG7$\"\"!1F/F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f*G6#% \"tG,$*&-%$sinGF&\"\"\",&F,!\"\"-%*HeavisideG6#,&%#PiGF.F'F,F,F,F." }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG,&*&\"\"\"F',&*$) %\"sG\"\"#F'F'F'F'!\"\"F'*&-%$expG6#,$*&F+F'%#PiGF'F-F'F(F-F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&,&\"\"\"F'-%$expG6#,$*&%\"sGF'%# PiGF'!\"\"F'F',&*$)F-\"\"#F'F'F'F'F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 36 " As before the Laplace transform of \+ " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 42 " is obtained b y dividing the transform of " }{XPPEDIT 18 0 "`f*`(t)" "6#-%#f*G6#%\"t G" }{TEXT -1 3 " by" }{XPPEDIT 18 0 "``(1-exp(-Pi*s));" "6#-%!G6#,&\" \"\"F'-%$expG6#,$*&%#PiGF'%\"sGF'!\"\"F/" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Fs/( 1-exp(-Pi*s));\n``=simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(, &\"\"\"F%-%$expG6#,$*&%\"sGF%%#PiGF%!\"\"F%F%,&*$)F+\"\"#F%F%F%F%F-,&F %F%F&F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*(,&\"\"\"F(-%$expG 6#,$*&%\"sGF(%#PiGF(!\"\"F(F(,&*$)F.\"\"#F(F(F(F(F0,&F(F0F)F(F0F0" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "We can also obtain this result \+ by using the integral formula:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[f(t)];" "6#*&%\"LG \"\"\"7#-%\"fG6#%\"tGF%" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "1/(1-exp(-p *s));" "6#*&\"\"\"F$,&F$F$-%$expG6#,$*&%\"pGF$%\"sGF$!\"\"F-F-" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(t)*exp(-t*s),t = 0 .. p);" "6#-%$ IntG6$*&-%\"fG6#%\"tG\"\"\"-%$expG6#,$*&F*F+%\"sGF+!\"\"F+/F*;\"\"!%\" pG" }{TEXT -1 4 " , " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " } {XPPEDIT 18 0 "f(t) = abs(sin(t));" "6#/-%\"fG6#%\"tG-%$absG6#-%$sinG6 #F'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "p = Pi;" "6#/%\"pG%#PiG" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 18 "Over the interval " } {XPPEDIT 18 0 "[0,Pi]" "6#7$\"\"!%#PiG" }{TEXT -1 10 ", we have " } {XPPEDIT 18 0 "sin*t>=0" "6#1\"\"!*&%$sinG\"\"\"%\"tGF'" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "f(t)=abs(sin*t)" "6#/-%\"fG6#%\"tG-%$absG 6#*&%$sinG\"\"\"F'F-" }{TEXT -1 20 " can be replaced by " }{XPPEDIT 18 0 "g(t)=sin*t" "6#/-%\"gG6#%\"tG*&%$sinG\"\"\"F'F*" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "g := t -> sin(t):\n'g(t)'=g(t);\nInt('g(t)'*exp(-s*t),t=0..Pi )/(1-exp(-Pi*s));\n``=value(%);\n``=simplify(map(expand,rhs(%)));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"tG-%$sinGF&" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,$-%$IntG6$*&-%\"gG6#%\"tG\"\"\"-%$expG6#,$*&%\" sGF,F+F,!\"\"F,/F+;\"\"!%#PiG*$,&F,F,-F.6#,$*&F2F,F7F,F3F3F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G**,&-%$expG6#*&%\"sG\"\"\"%#PiGF,F,F,F,F ,-F(6#,$F*!\"\"F,,&*$)F+\"\"#F,F,F,F,F1,&F,F,F.F1F1" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%!G*(,&-%$expG6#*&%\"sG\"\"\"%#PiGF,F,F,F,F,,&*$)F+ \"\"#F,F,F,F,!\"\",&F'F,F,F2F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 7 "laplace" } {TEXT -1 48 " gives this result when applied to the function " } {XPPEDIT 18 0 "abs(sin*t);" "6#-%$absG6#*&%$sinG\"\"\"%\"tGF(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "abs(sin(t));\n`Laplace transform`=inttrans[laplace](% ,t,s);\n``=simplify(convert(rhs(%),exp));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$absG6#-%$sinG6#%\"tG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG*&,&*$)%\"sG\"\"#\"\"\"F+F+F+!\"\"-%%cothG6 #,$*(F*F,F)F+%#PiGF+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*(,&-% $expG6#*&%\"sG\"\"\"%#PiGF,F,F,F,F,,&*$)F+\"\"#F,F,F,F,!\"\",&F'F,F,F2 F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 74 "Example 5: Laplace transform of the half-wave rectification of a sine wave" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "f(t) = PIECEWISE([sin*t, 0 <= sin*t],[0, sin*t < 0]);" "6#/-%\"fG6#%\"tG-%*PIECEWISEG6$7$*&%$sinG\"\"\"F'F.1\"\"!*&F-F.F'F.7$ F02*&F-F.F'F.F0" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(sin*t+abs(sin*t)) /2" "6#/%!G*&,&*&%$sinG\"\"\"%\"tGF)F)-%$absG6#*&F(F)F*F)F)F)\"\"#!\" \"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "plot((sin(t)+abs(sin(t)))/2,t=0..16.5,-0.2 ..1.2,thickness=2,\n numpoints=100,color=COLOR(RGB,.5,0,.7),labels=[ t,`f(t)`],\n ytickmarks=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 521 152 152 {PLOTDATA 2 "6(-%'CURVESG6#7bv7$$\"\"!F)F(7$$\"3Hmmmm;')=()!#>$\"3 j#G)yx#>yq)F-7$$\"3ELLLLBxV5E$F3$\"3(px\"H\"QGN?$F37$$\"31+++v2<9TF3$\"3 c+!pge&3**RF37$$\"3MLLLLAKn\\F3$\"3kI<^,CblZF37$$\"3ELLLL*Gh#eF3$\"3s: *)\\7l2-bF37$$\"3=LLLLc$\\o'F3$\"3OI'z;0X!)>'F37$$\"34+++v0mRvF3$\"3u' 4@f6``%oF37$$\"3)emmm^&Q%R)F3$\"3vCkMnDoUuF37$$\"3w)*****\\\\#o=*F3$\" 3)QF:neF![zF37$$\"3wKLL$Qk#z**F3$\"3>(=GWE)[.%)F37$$\"3))*****\\YJ?;\" !#<$\"3sj1P2Y7w\"*F37$$\"3?LLL=\"\\Y@e%**F37$$\"31+++0Yd^3]dAFeo$\"3m6]1e`RLxF37$$\"3)*******RP)4M#Feo$\"3!o2w1#>!y<(F37 $$\"3Umm;HUz;CFeo$\"3yK5*)*\\X)HmF37$$\"3ILLL=Zg#\\#Feo$\"3.c_*e2/Q/'F 37$$\"3;++]A2v#e#Feo$\"3<`)R9iY?I&F37$$\"3cmmmEn*Gn#Feo$\"3fL83$Feo$\"3)H*eca@kjfF-7$$\"3Immm1:bgJFeoF(7$$\"3-LL $e#=#oC$FeoF(7$$\"3<+++X@4LLFeoF(7$$\"31+++N;R(\\$FeoF(7$$\"3wmmm;4#)o OFeoF(7$$\"3jmmm6lCEQFeoF(7$$\"3ELLL$G^g*RFeoF(7$$\"3oKLL=2VsTFeoF(7$$ \"3f*****\\`pfK%FeoF(7$$\"3!HLLLm&z\"\\%FeoF(7$$\"3s******z-6jYFeoF(7$ $\"3<******4#32$[FeoF(7$$\"3O*****\\#y'G*\\FeoF(7$$\"3G******H%=H<&Feo F(7$$\"35mmm1>qM`FeoF(7$$\"3%)*******HSu]&FeoF(7$$\"3'HLL$ep'Rm&FeoF(7 $$\"3')******R>4NeFeoF(7$$\"3#emm;@2h*fFeoF(7$$\"37LLL))3E!3'FeoF(7$$ \"3]*****\\c9W;'FeoF(7$$\"3mmmTlBb0iFeoF(7$$\"3#HLLe;!pYiFeoF(7$$\"3gl ;/m!fsE'FeoF(7$$\"3=***\\i'z#yG'Feo$\"3?vDB#RUEk%!#?7$$\"3vK$ek'oR3jFe o$\"3#*HTO_)))3_#F-7$$\"3Lmmmmd'*GjFeo$\"3Kb!>j'pWwXF-7$$\"3)HLLep+^T' Feo$\"3%)H\"R1IJ`J\"F37$$\"3j*****\\iN7]'Feo$\"3s)Qlk>lK;#F37$$\"39mm; 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" }}{PARA 0 "" 0 "" {TEXT -1 60 "The \+ associated (non-periodic) function which coincides with " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 38 " over the \"first period\" , that is for " }{XPPEDIT 18 0 "0<=t" "6#1\"\"!%\"tG" }{XPPEDIT 18 0 " `` < 2*Pi;" "6#2%!G*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 30 ", and is zero \+ thereafter is: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f*`(t) = PIECEWISE([sin*t, 0 <= t and \+ t < Pi],[0, Pi <= t]);" "6#/-%#f*G6#%\"tG-%*PIECEWISEG6$7$*&%$sinG\"\" \"F'F.31\"\"!F'2F'%#PiG7$F11F3F'" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = \+ sin*t-u[Pi](t)*sin*t;" "6#/%!G,&*&%$sinG\"\"\"%\"tGF(F(*(-&%\"uG6#%#Pi G6#F)F(F'F(F)F(!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "This is the same " }{XPPEDIT 18 0 "`f*` (t)" "6#-%#f*G6#%\"tG" }{TEXT -1 39 " as for the full wave rectificati on of " }{XPPEDIT 18 0 "sin*t;" "6#*&%$sinG\"\"\"%\"tGF%" }{TEXT -1 36 " considered in the previous example." }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 1 "u" }{TEXT -1 4 " .. " }{HYPERLNK 17 "u " 1 "" "u" }{TEXT -1 88 " from the subsection which follows the exampl es, and which implements the step function " }{XPPEDIT 18 0 "u[a](t)=` ` " "6#/-&%\"uG6#%\"aG6#%\"tG%!G" }{TEXT 297 7 "u[a](t)" }{TEXT -1 42 " may be used to set up the Maple function " }{TEXT 0 4 "`f*`" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "`f*`(t)" "6#-%#f*G6#%\"tG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 206 "`f*` := t -> sin(t)-'u[Pi]'(t)*sin(t):\n'`f*`(t)'=`f *`(t);\n``=convert(rhs(%),piecewise);\nplot(`f*`(t),t=0..8.5,0..1.2,th ickness=2,color=COLOR(RGB,1,.15,0),\n labels=[t,`f*(t)`],ytickmark s=3,numpoints=75);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f*G6#%\"tG,& -%$sinGF&\"\"\"*&-&%\"uG6#%#PiGF&F+F)F+!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%*PIECEWISEG6$7$-%$sinG6#%\"tG2F,%#PiG7$\"\"!1F.F, " }}{PARA 13 "" 1 "" {GLPLOT2D 559 147 147 {PLOTDATA 2 "6(-%'CURVESG6# 7\\q7$$\"\"!F)F(7$$\"+0_%*3g!#6$\"+K(H`+'F-7$$\"+T!*y,7!#5$\"+H#)*))> \"F37$$\"+]TiC&F3$\"+H*[b'\\F37$$\"+](>`y&F3$\"+')R&zY&F37$$ \"+S/YJjF3$\"+$GRo\"fF37$$\"+J6gxoF3$\"+I33[jF37$$\"+M_f3!)F3$\"+;mazr F37$$\"+`4Fy\"*F3$\"+MN$G%zF37$$\"+^&>W.\"!\"*$\"+u(ecf)F37$$\"+A#\\V: \"F_p$\"+W!>`9*F37$$\"+UL)*f7F_p$\"+sC&3_*F37$$\"+UB!*y8F_p$\"+\"yZk\" )*F37$$\"+))fgQ9F_p$\"+Fdv7**F37$$\"+M'4$)\\\"F_p$\"+)*)RP(**F37$$\"+( Rxq_\"F_p$\"+)zW/***F37$$\"+g^%eb\"F_p$\"+MB)))***F37$$\"+BHh%e\"F_p$ \"+Cb/****F37$$\"+'o!Q8;F_p$\"+KU$4***F37$$\"+c%Gcm\"F_p$\"+D!o]&**F37 $$\"+Di(yr\"F_p$\"+jB.#*)*F37$$\"+wQ+!y\"F_p$\"+=!e>y*F37$$\"+G:8U=F_p $\"+s*QTj*F37$$\"+p?RZ>F_p$\"+4'>#*H*F37$$\"+,^$)p?F_p$\"+E*H/y)F37$$ \"+Jz@y@F_p$\"+2PA6#)F37$$\"+u58(H#F_p$\"+tj8wuF37$$\"+1XO5CF_p$\"+gJ% yn'F37$$\"+9<^GDF_p$\"+L-!Rv&F37$$\"+B(fFe#F_p$\"+4.(>I&F37$$\"+Kx+PEF _p$\"+E8WM[F37$$\"+Q:_&p#F_p$\"+NVC9VF37$$\"+X`.aFF_p$\"+d,GzPF37$$\"+ XZ\"[\"GF_p$\"+n6$*4KF37$$\"+XTfvGF_p$\"+1zsGEF37$$\"+\"z-&GHF_p$\"+6) 3[6#F37$$\"+P9T\")HF_p$\"+g5(\\f\"F37$$\"+CUbQIF_p$\"+8?cG5F37$$\"+7qp &4$F_p$\"+yI&ze%F-7$$\"+eS@DJF_p$\"+KjyP;F-7$$\"+.6taJF_pF(7$$\"+[\"[U =$F_pF(7$$\"+%>lP@$F_pF(7$$\"+M&=:F$F_pF(7$$\"+v=FHLF_pF(7$$\"+H0.TMF_ pF(7$$\"+g%>^c$F_pF(7$$\"+$))=mn$F_pF(7$$\"+;$ocz$F_pF(7$$\"+(zWN!RF_p F(7$$\"+%G#[@SF_pF(7$$\"+pAXKTF_pF(7$$\"+xyW[UF_pF(7$$\"+'ya=O%F_pF(7$ $\"+58e![%F_pF(7$$\"+k!H\\f%F_pF(7$$\"+eh'=r%F_pF(7$$\"+`\\$y#[F_pF(7$ $\"+K#)RM\\F_pF(7$$\"+w<`c]F_pF(7$$\"+t%pd;&F_pF(7$$\"+U'RAG&F_pF(7$$ \"+bkr$R&F_pF(7$$\"+Ir\\_\\yfF_pF(7$$\"+y-9$3'F_pF(7$$\"+&y=B?'F_pF(7$$\"+R$ RuJ'F_pF(7$$\"+)o%[KkF_pF(7$$\"+0o5ZlF_pF(7$$\"+wFAdmF_pF(7$$\"+d5EwnF _pF(7$$\"+rJr*)oF_pF(7$$\"+xh64qF_pF(7$$\"+_Y?v#)F_pF(7$$\"+/Gn#Q)F_pF(7$$\"#&)!\"\"F(-%+AXESLABELSG6$%\"tG%&f *(t)G-%&COLORG6&%$RGBG$\"\"\"F)$\"#:!\"#F(-%*THICKNESSG6#\"\"#-%*AXEST ICKSG6$%(DEFAULTG\"\"$-%%VIEWG6$;F(F\\cl;F($\"#7F^cl" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "The Laplace transform of " }{XPPEDIT 18 0 "`f*` (t)" "6#-%#f*G6#%\"tG" }{TEXT -1 60 " can be obtained by making use of the second shift formula: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "L*[u[a](t)*g(t)] = exp(-a*s)*L*[g(t+a)];" "6#/*&%\"LG\" \"\"7#*&-&%\"uG6#%\"aG6#%\"tGF&-%\"gG6#F/F&F&*(-%$expG6#,$*&F-F&%\"sGF &!\"\"F&F%F&7#-F16#,&F/F&F-F&F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 8 "We have " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[`f*`(t)] = L*[sin(t)-u[Pi](t)*s in(t)];" "6#/*&%\"LG\"\"\"7#-%#f*G6#%\"tGF&*&F%F&7#,&-%$sinG6#F+F&*&-& %\"uG6#%#PiG6#F+F&-F06#F+F&!\"\"F&" }{TEXT -1 1 " " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(s^2+1)-exp(-Pi*s)*L*[sin(t+Pi )];" "6#/%!G,&*&\"\"\"F',&*$%\"sG\"\"#F'F'F'!\"\"F'*(-%$expG6#,$*&%#Pi GF'F*F'F,F'%\"LGF'7#-%$sinG6#,&%\"tGF'F3F'F'F," }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = 1/(s^2+1)+exp(-Pi*s)*L*[sin(t)];" "6#/%!G,&*&\"\" \"F',&*$%\"sG\"\"#F'F'F'!\"\"F'*(-%$expG6#,$*&%#PiGF'F*F'F,F'%\"LGF'7# -%$sinG6#%\"tGF'F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " \+ " }{XPPEDIT 18 0 "`` = 1/(s^2+1)+exp(-Pi*s)/(s^2+1);" "6#/%!G,&*&\"\" \"F',&*$%\"sG\"\"#F'F'F'!\"\"F'*&-%$expG6#,$*&%#PiGF'F*F'F,F',&*$F*F+F 'F'F'F,F'" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = (1+exp(-Pi*s))/(s^2+1 );" "6#/%!G*&,&\"\"\"F'-%$expG6#,$*&%#PiGF'%\"sGF'!\"\"F'F',&*$F.\"\"# F'F'F'F/" }{TEXT -1 30 ", as in the previous example. " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" } {TEXT -1 12 " has period " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiG F%" }{TEXT -1 19 ", it follows that " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "L*[f(t)] = 1/(1-exp(-2*Pi*s));" "6#/*&%\"LG\"\"\" 7#-%\"fG6#%\"tGF&*&F&F&,&F&F&-%$expG6#,$*(\"\"#F&%#PiGF&%\"sGF&!\"\"F6 F6" }{XPPEDIT 18 0 "``((1+exp(-Pi*s))/(s^2+1));" "6#-%!G6#*&,&\"\"\"F( -%$expG6#,$*&%#PiGF(%\"sGF(!\"\"F(F(,&*$F/\"\"#F(F(F(F0" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (1+exp(-Pi* s))/((1+exp(-Pi*s))*(1-exp(-Pi*s))*(s^2+1));" "6#/%!G*&,&\"\"\"F'-%$ex pG6#,$*&%#PiGF'%\"sGF'!\"\"F'F'*(,&F'F'-F)6#,$*&F-F'F.F'F/F'F',&F'F'-F )6#,$*&F-F'F.F'F/F/F',&*$F.\"\"#F'F'F'F'F/" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/((1-exp(-Pi*s))*(s^2+1));" "6#/%!G*&\"\"\"F&*&,&F&F&-%$expG6 #,$*&%#PiGF&%\"sGF&!\"\"F0F&,&*$F/\"\"#F&F&F&F&F0" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "The Lapla ce transform of " }{XPPEDIT 18 0 "`f*`(t)" "6#-%#f*G6#%\"tG" }{TEXT -1 25 " can be determined using " }{TEXT 0 7 "laplace" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 210 "`f*` := t -> sin(t)-'u[Pi]'(t)*sin(t):\n'`f*`(t)'=`f*`(t);\nc onvert(rhs(%),Heaviside):\n`f*` := unapply(%,t):\n'`f*`(t)'=`f*`(t);\n `Laplace transform`=inttrans[laplace](`f*`(t),t,s);\n``=normal(rhs(%)) ;\nFs := rhs(%):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f*G6#%\"tG,&-% $sinGF&\"\"\"*&-&%\"uG6#%#PiGF&F+F)F+!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f*G6#%\"tG,&-%$sinGF&\"\"\"*&-%*HeavisideG6#,&%#PiG !\"\"F'F+F+F)F+F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transf ormG,&*&\"\"\"F',&*$)%\"sG\"\"#F'F'F'F'!\"\"F'*&-%$expG6#,$*&F+F'%#PiG F'F-F'F(F-F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&,&\"\"\"F'-%$exp G6#,$*&%\"sGF'%#PiGF'!\"\"F'F',&*$)F-\"\"#F'F'F'F'F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Since the periodic fun ction " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 12 " has per iod " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 27 ", t he Laplace transform of " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" } {TEXT -1 42 " is obtained by dividing the transform of " }{XPPEDIT 18 0 "`f*`(t)" "6#-%#f*G6#%\"tG" }{TEXT -1 3 " by" }{XPPEDIT 18 0 "``(1-e xp(-2*Pi*s));" "6#-%!G6#,&\"\"\"F'-%$expG6#,$*(\"\"#F'%#PiGF'%\"sGF'! \"\"F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Fs/(1-exp(-2*Pi*s));\n``=simplify(% );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(,&\"\"\"F%-%$expG6#,$*&%\"sGF% %#PiGF%!\"\"F%F%,&*$)F+\"\"#F%F%F%F%F-,&F%F%-F'6#,$*(F1F%F+F%F,F%F-F-F -" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&\"\"\"F'*&,&-%$expG6#,$*& %\"sGF'%#PiGF'!\"\"F'F'F1F',&*$)F/\"\"#F'F'F'F'F'F1F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Alternatively, we can start with the pi ecewise definition of " }{XPPEDIT 18 0 "`f*`(t)" "6#-%#f*G6#%\"tG" } {TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f* `(t) = PIECEWISE([sin*t, 0 <= t and t < Pi],[0, Pi <= t]);" "6#/-%#f*G 6#%\"tG-%*PIECEWISEG6$7$*&%$sinG\"\"\"F'F.31\"\"!F'2F'%#PiG7$F11F3F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 227 "`f*` := t -> piecewise(t=Pi,0): \n'`f*`(t)'=`f*`(t);\nsimplify(convert(`f*`(t),Heaviside)):\n`f*` := u napply(%,t):\n'`f*`(t)'=`f*`(t);\n`Laplace transform`=inttrans[laplace ](`f*`(t),t,s);\n``=normal(rhs(%));\nFs := rhs(%):" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%#f*G6#%\"tG-%*PIECEWISEG6$7$-%$sinGF&2F'%#PiG7$\" \"!1F/F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f*G6#%\"tG,$*&-%$sinGF &\"\"\",&F,!\"\"-%*HeavisideG6#,&%#PiGF.F'F,F,F,F." }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%2Laplace~transformG,&*&\"\"\"F',&*$)%\"sG\"\"#F'F'F 'F'!\"\"F'*&-%$expG6#,$*&F+F'%#PiGF'F-F'F(F-F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&,&\"\"\"F'-%$expG6#,$*&%\"sGF'%#PiGF'!\"\"F'F',&* $)F-\"\"#F'F'F'F'F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 36 " As before the Laplace transform of " }{XPPEDIT 18 0 "f (t)" "6#-%\"fG6#%\"tG" }{TEXT -1 42 " is obtained by dividing the tran sform of " }{XPPEDIT 18 0 "`f*`(t)" "6#-%#f*G6#%\"tG" }{TEXT -1 3 " by " }{XPPEDIT 18 0 "``(1-exp(-2*Pi*s));" "6#-%!G6#,&\"\"\"F'-%$expG6#,$* (\"\"#F'%#PiGF'%\"sGF'!\"\"F0" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Fs/(1-exp(-2 *Pi*s));\n``=simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(,&\"\" \"F%-%$expG6#,$*&%\"sGF%%#PiGF%!\"\"F%F%,&*$)F+\"\"#F%F%F%F%F-,&F%F%-F '6#,$*(F1F%F+F%F,F%F-F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*& \"\"\"F'*&,&*$)%\"sG\"\"#F'F'F'F'F',&F'!\"\"-%$expG6#,$*&F,F'%#PiGF'F/ F'F'F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "We can also obtain this result by using the integral formula: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[f(t)];" "6#*&%\"LG\"\"\"7#-%\"fG6#% \"tGF%" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "1/(1-exp(-p*s));" "6#*&\"\" \"F$,&F$F$-%$expG6#,$*&%\"pGF$%\"sGF$!\"\"F-F-" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int(f(t)*exp(-t*s),t = 0 .. p);" "6#-%$IntG6$*&-%\"fG6# %\"tG\"\"\"-%$expG6#,$*&F*F+%\"sGF+!\"\"F+/F*;\"\"!%\"pG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "whe re " }{XPPEDIT 18 0 "p=2*Pi" "6#/%\"pG*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "f(t)=(sin*t+abs(sin*t))/2" "6#/-%\"fG6#%\"tG *&,&*&%$sinG\"\"\"F'F,F,-%$absG6#*&F+F,F'F,F,F,\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = PIECEWISE([sin*t, 0 <= sin*t],[0, sin*t < 0 ]);" "6#/%!G-%*PIECEWISEG6$7$*&%$sinG\"\"\"%\"tGF+1\"\"!*&F*F+F,F+7$F. 2*&F*F+F,F+F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 8 "We have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(t)*exp(-t*s),t = 0 .. 2*Pi) = Int(exp(-t*s)*s in*t,t = 0 .. Pi);" "6#/-%$IntG6$*&-%\"fG6#%\"tG\"\"\"-%$expG6#,$*&F+F ,%\"sGF,!\"\"F,/F+;\"\"!*&\"\"#F,%#PiGF,-F%6$*(-F.6#,$*&F+F,F2F,F3F,%$ sinGF,F+F,/F+;F6F9" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "Int(sin(t)*exp(-s*t),t=0..P i)/(1-exp(-2*Pi*s));\n``=value(%);\n``=simplify(map(expand,rhs(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$IntG6$*&-%$sinG6#%\"tG\"\"\"-%$ expG6#,$*&%\"sGF,F+F,!\"\"F,/F+;\"\"!%#PiG*$,&F,F,-F.6#,$*(\"\"#F,F2F, F7F,F3F3F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G**,&-%$expG6#*&%\"sG \"\"\"%#PiGF,F,F,F,F,-F(6#,$F*!\"\"F,,&*$)F+\"\"#F,F,F,F,F1,&F,F,-F(6# ,$*(F5F,F+F,F-F,F1F1F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*(-%$exp G6#*&%\"sG\"\"\"%#PiGF+F+,&F&F+F+!\"\"F.,&*$)F*\"\"#F+F+F+F+F." }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The proce dure " }{TEXT 0 7 "laplace" }{TEXT -1 48 " gives this result when appl ied to the function " }{XPPEDIT 18 0 "(sin(t)+abs(sin(t)))/2;" "6#*&,& -%$sinG6#%\"tG\"\"\"-%$absG6#-F&6#F(F)F)\"\"#!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "(sin(t)+abs(sin(t)))/2;\n`Laplace transform`=inttrans[laplace](%, t,s);\n``=simplify(convert(rhs(%),exp));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&#\"\"\"\"\"#F&-%$sinG6#%\"tGF&F&*&F%F&-%$absG6#F(F&F&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG,&*&\"\"\"F'*&\" \"#F',&*$)%\"sGF)F'F'F'F'F'!\"\"F'*&#F'F)F'*&F*F.-%%cothG6#,$*(F)F.F-F '%#PiGF'F'F'F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*(-%$expG6#*&% \"sG\"\"\"%#PiGF+F+,&F&F+F+!\"\"F.,&*$)F*\"\"#F+F+F+F+F." }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Example 6: Laplace transform of \+ a parabolic wave" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(t) = PIECEWISE([2*t-t^2, 0 <= t and \+ t < 2],[t^2-6*t+8, 2 <= t and t < 4]);" "6#/-%\"fG6#%\"tG-%*PIECEWISEG 6$7$,&*&\"\"#\"\"\"F'F/F/*$F'F.!\"\"31\"\"!F'2F'F.7$,(*$F'F.F/*&\"\"'F /F'F/F1\"\")F/31F.F'2F'\"\"%" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 " and " }{XPPEDIT 18 0 "f (t);" "6#-%\"fG6#%\"tG" }{TEXT -1 28 " is periodic with period 4. 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" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The proc edure " }{TEXT 0 1 "u" }{TEXT -1 4 " .. " }{HYPERLNK 17 "u" 1 "" "u" } {TEXT -1 88 " from the subsection which follows the examples, and whic h implements the step function " }{XPPEDIT 18 0 "u[a](t)=`` " "6#/-&% \"uG6#%\"aG6#%\"tG%!G" }{TEXT 297 7 "u[a](t)" }{TEXT -1 42 " may be us ed to set up the Maple function " }{TEXT 0 4 "`f*`" }{TEXT -1 5 " for \+ " }{XPPEDIT 18 0 "`f*`(t)" "6#-%#f*G6#%\"tG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 226 "` f*` := t -> 2*t-t^2+'u[2]'(t)*(2*t^2-8*t+8)-'u[4]'(t)*(t^2-6*t+8):\n'` f*`(t)'=`f*`(t);\n``=convert(rhs(%),piecewise);\nplot(`f*`(t),t=0..8.5 ,-1.2..1.2,thickness=2,color=COLOR(RGB,.2,.7,.3),\n labels=[t,`f*( t)`],ytickmarks=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f*G6#%\"tG, **&\"\"#\"\"\"F'F+F+*$)F'F*F+!\"\"*&-&%\"uG6#F*F&F+,(*&F*F+F-F+F+*&\" \")F+F'F+F.F7F+F+F+*&-&F26#\"\"%F&F+,(F,F+*&\"\"'F+F'F+F.F7F+F+F." }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%*PIECEWISEG6%7$,&*&\"\"#\"\"\"% \"tGF,F,*$)F-F+F,!\"\"2F-F+7$,(F.F,*&\"\"'F,F-F,F0\"\")F,2F-\"\"%7$\" \"!1F8F-" }}{PARA 13 "" 1 "" {GLPLOT2D 412 189 189 {PLOTDATA 2 "6(-%'C URVESG6#7ep7$$\"\"!F)F(7$$\"+?0zj#*!#6$\"+*GSpw\"!#57$$\"+/\"eF&=F0$\" +\\\\CiLF07$$\"+kbzeEF0$\"+VE!>'F0$\"+I'*e[&)F07 $$\"+4Tu-rF0$\"+K3fg\"*F07$$\"+g$*)3,)F0$\"+')QM/'*F07$$\"+5Y.>*)F0$\" +$Q^J))*F07$$\"+g!=+M*F0$\"+%QUk&**F07$$\"+5:+h(*F0$\"+szG%***F07$$\"+ NK\\r**F0$\"+O(=*****F07$$\"+'\\)>=5!\"*$\"*:)o'***Fjo7$$\"+owCR5Fjo$ \"*@'f%)**Fjo7$$\"+ToHg5Fjo$\"*\"Hkj**Fjo7$$\"+OwZZ6Fjo$\"*Z.Dy*Fjo7$$ \"+J%eYB\"Fjo$\"*@a$\\%*Fjo7$$\"+M4#[K\"Fjo$\"*h8\\%*)Fjo7$$\"+QM)\\T \"Fjo$\"*Y()yF)Fjo7$$\"+*yc[]\"Fjo$\"*B'>^uFjo7$$\"+S,t%f\"Fjo$\"*hgHY 'Fjo7$$\"+Gf<(o\"Fjo$\"*W#*yF&Fjo7$$\"+<Fjo$\"*g@u6\"Fjo7$$\"+Z59M?Fjo$!)([;r'Fjo7 $$\"+wx!e7#Fjo$!*ezyN#Fjo7$$\"+(*3&y@#Fjo$!*&yU#)QFjo7$$\"+=S*)4BFjo$! *GPvB&Fjo7$$\"+?Yf)R#Fjo$!*r:JQ'Fjo7$$\"+B_H([#Fjo$!*5Q8P(Fjo7$$\"+oQ% yc#Fjo$!*u5C8)Fjo7$$\"+8DR[EFjo$!*n@Pw)Fjo7$$\"+UE_!)GFjo$!**)\\s&)*Fjo7$$\"+064@HFjo$!* %QtP**Fjo7$$\"+s-mhHFjo$!*t+`)**Fjo7$$\"+Q%HA+$Fjo$!*A]*****Fjo7$$\"+I _#e-$Fjo$!*iIL***Fjo7$$\"+B5U\\IFjo$!*gvb(**Fjo7$$\"+;o,tIFjo$!*\\&oY* *Fjo7$$\"+3Eh'4$Fjo$!*)*fm!**Fjo7$$\"+$>/Q9$Fjo$!*[.Kz*Fjo7$$\"+yd*4>$ Fjo$!*/1_j*Fjo7$$\"+w3auKFjo$!*3tiC*Fjo7$$\"+vf3eLFjo$!*IWxr)Fjo7$$\"+ *=[(\\MFjo$!*ils(zFjo7$$\"+./TTNFjo$!*uZ(oqFjo7$$\"+2XpGOFjo$!*;Ku/'Fj o7$$\"+6'yfr$Fjo$!*GYP([Fjo7$$\"+Fjo7$$\"+'RQ*RRFjo$!*=e^;\"Fjo7$$\"+qXv\")RFjo$!)gz:OFjo7$$\"+9' 3A*RFjo$!)i?_:Fjo7$$\"+dEm-SFjo$!\"'Fjo7$$\"++n68SFjoF(7$$\"+W2dBSFjo$ !\"#Fjo7$$\"+J)yW/%Fjo$\"\"#Fjo7$$\"+=pQlSFjo$\"\"'Fjo7$$\"+.dfbTFjoFi ^l7$$\"+)[/eC%FjoFi^l7$$\"+Pw?LWFjo$!\"\"!\")7$$\"+IEM'f%FjoF(7$$\"+\" *G`sZFjoF(7$$\"+sYba\\FjoFd_l7$$\"+suiK^FjoFd_l7$$\"+j?#\\I&Fjo$\"\"\" Ff_l7$$\"+JeA'\\&FjoF(7$$\"+v27ocFjoF(7$$\"+=`l^eFjoFf`l7$$\"+#*['z,'F joFd_l7$$\"+&=&y*>'FjoF(7$$\"+6R'3P'FjoF(7$$\"+u/p\\lFjoF(7$$\"+Eh_CnF joF(7$$\"+]Gc2pFjoF(7$$\"+q*[Q3(Fjo$\"\"%Ff_l7$$\"+!*p7ksFjoFfbl7$$\"+ ;A\"HW(Fjo$F`^lFf_l7$$\"+rs>2wFjo$!\"%Ff_l7$$\"+%['[&z(FjoFbcl7$$\"++Y *Q'zFjoF(7$$\"+yFXV\")FjoFf`l7$$\"+kGJ:$)FjoFfbl7$$\"#&)Fe_lF(-%+AXESL ABELSG6$%\"tG%&f*(t)G-%&COLORG6&%$RGBG$Fe^lFe_l$\"\"(Fe_l$\"\"$Fe_l-%* THICKNESSG6#Fe^l-%*AXESTICKSG6$%(DEFAULTGF`el-%%VIEWG6$;F(Fadl;$!#7Fe_ l$\"#7Fe_l" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "C urve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "The Laplace transfor m of " }{XPPEDIT 18 0 "`f*`(t)" "6#-%#f*G6#%\"tG" }{TEXT -1 60 " can b e obtained by making use of the second shift formula: " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[u[a](t)*g(t)] = exp(-a*s)*L*[ g(t+a)];" "6#/*&%\"LG\"\"\"7#*&-&%\"uG6#%\"aG6#%\"tGF&-%\"gG6#F/F&F&*( -%$expG6#,$*&F-F&%\"sGF&!\"\"F&F%F&7#-F16#,&F/F&F-F&F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 8 "We have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[`f*`(t)] = L*[2*t-t^2+u[2](t)*(2*t^2-8*t+8 )-u[4](t)*(t^2-6*t+8)];" "6#/*&%\"LG\"\"\"7#-%#f*G6#%\"tGF&*&F%F&7#,** &\"\"#F&F+F&F&*$F+F0!\"\"*&-&%\"uG6#F06#F+F&,(*&F0F&*$F+F0F&F&*&\"\")F &F+F&F2F=F&F&F&*&-&F66#\"\"%6#F+F&,(*$F+F0F&*&\"\"'F&F+F&F2F=F&F&F2F& " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2/(s^2)-2/(t^3)+exp(-2*s)*L*[2*(t+ 2)^2-8*(t+2)+8]-exp(-4*s)*L*[(t+4)^2-6*(t+4)+8];" "6#/%!G,**&\"\"#\"\" \"*$%\"sGF'!\"\"F(*&F'F(*$%\"tG\"\"$F+F+*(-%$expG6#,$*&F'F(F*F(F+F(%\" LGF(7#,(*&F'F(*$,&F.F(F'F(F'F(F(*&\"\")F(,&F.F(F'F(F(F+F=F(F(F(*(-F26# ,$*&\"\"%F(F*F(F+F(F6F(7#,(*$,&F.F(FDF(F'F(*&\"\"'F(,&F.F(FDF(F(F+F=F( F(F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2/(s^2)-2/(s^3)+exp(-2*s)*L*[ 2*t^2]-exp(-4*s)*L*[t^2+2*t];" "6#/%!G,**&\"\"#\"\"\"*$%\"sGF'!\"\"F(* &F'F(*$F*\"\"$F+F+*(-%$expG6#,$*&F'F(F*F(F+F(%\"LGF(7#*&F'F(*$%\"tGF'F (F(F(*(-F16#,$*&\"\"%F(F*F(F+F(F5F(7#,&*$F9F'F(*&F'F(F9F(F(F(F+" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2/(s^2)-2/(s^3)+4*exp(-2*s)/(s^3)-exp(-4*s)*(2/(s^3)+2/(s^2));" "6#/% !G,**&\"\"#\"\"\"*$%\"sGF'!\"\"F(*&F'F(*$F*\"\"$F+F+*(\"\"%F(-%$expG6# ,$*&F'F(F*F(F+F(*$F*F.F+F(*&-F26#,$*&F0F(F*F(F+F(,&*&F'F(*$F*F.F+F(*&F 'F(*$F*F'F+F(F(F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 32 " has period 4 , it follows that: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[f(t)] = 1/(1-exp(-4*s));" "6#/*&%\"LG\"\"\"7#-%\"fG6#%\"tGF&*&F& F&,&F&F&-%$expG6#,$*&\"\"%F&%\"sGF&!\"\"F5F5" }{XPPEDIT 18 0 "``(2/(s^ 2)-2/(s^3)+4*exp(-2*s)/(s^3)-exp(-4*s)*(2/(s^3)+2/(s^2)));" "6#-%!G6#, **&\"\"#\"\"\"*$%\"sGF(!\"\"F)*&F(F)*$F+\"\"$F,F,*(\"\"%F)-%$expG6#,$* &F(F)F+F)F,F)*$F+F/F,F)*&-F36#,$*&F1F)F+F)F,F),&*&F(F)*$F+F/F,F)*&F(F) *$F+F(F,F)F)F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2*(s-1+2*exp(-2* s)-exp(-4*s)*s-exp(-4*s))/(s^3*(1-exp(-4*s)));" "6#/%!G*(\"\"#\"\"\",, %\"sGF'F'!\"\"*&F&F'-%$expG6#,$*&F&F'F)F'F*F'F'*&-F-6#,$*&\"\"%F'F)F'F *F'F)F'F*-F-6#,$*&F6F'F)F'F*F*F'*&F)\"\"$,&F'F'-F-6#,$*&F6F'F)F'F*F*F' F*" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2*(1-exp(-2*s))*(exp(-2*s)*s+ex p(-2*s)-1+s)/(s^3*(1-exp(-4*s)));" "6#/%!G**\"\"#\"\"\",&F'F'-%$expG6# ,$*&F&F'%\"sGF'!\"\"F/F',**&-F*6#,$*&F&F'F.F'F/F'F.F'F'-F*6#,$*&F&F'F. F'F/F'F'F/F.F'F'*&F.\"\"$,&F'F'-F*6#,$*&\"\"%F'F.F'F/F/F'F/" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2*(exp(-2*s)*s+exp(-2*s)-1+s)/(s^3*(1+ex p(-2*s)));" "6#/%!G*(\"\"#\"\"\",**&-%$expG6#,$*&F&F'%\"sGF'!\"\"F'F/F 'F'-F+6#,$*&F&F'F/F'F0F'F'F0F/F'F'*&F/\"\"$,&F'F'-F+6#,$*&F&F'F/F'F0F' F'F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "The Laplace transform of " }{XPPEDIT 18 0 "`f*`(t)" "6 #-%#f*G6#%\"tG" }{TEXT -1 25 " can be determined using " }{TEXT 0 7 "l aplace" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 239 "`f*` := t -> 2*t-t^2+'u[2]'(t)*(2* t^2-8*t+8)-'u[4]'(t)*(t^2-6*t+8):\n'`f*`(t)'=`f*`(t);\nconvert(rhs(%), Heaviside):\n`f*` := unapply(%,t):\n'`f*`(t)'=`f*`(t);\n`Laplace trans form`=inttrans[laplace](`f*`(t),t,s);\n``=normal(rhs(%));\nFs := rhs(% ):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f*G6#%\"tG,**&\"\"#\"\"\"F'F +F+*$)F'F*F+!\"\"*&-&%\"uG6#F*F&F+,(*&F*F+F-F+F+*&\"\")F+F'F+F.F7F+F+F +*&-&F26#\"\"%F&F+,(F,F+*&\"\"'F+F'F+F.F7F+F+F." }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%#f*G6#%\"tG,2*&\"\"#\"\"\"F'F+F+*$)F'F*F+!\"\"*(F*F+ -%*HeavisideG6#,&F*F.F'F+F+F-F+F+*(\"\")F+F0F+F'F+F.*&F5F+F0F+F+*&-F16 #,&F'F+\"\"%F.F+F-F+F.*(\"\"'F+F8F+F'F+F+*&F5F+F8F+F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG,,*&\"\"#\"\"\"%\"sG!\"#F(*&F 'F(F)!\"$!\"\"*(\"\"%F(-%$expG6#,$*&F'F(F)F(F-F(F)F,F(*(F'F(-F16#,$*&F /F(F)F(F-F(F)F*F-*(F'F(F6F(F)F,F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %!G,$*(\"\"#\"\"\",,%\"sG!\"\"F(F(*&F'F(-%$expG6#,$*&F'F(F*F(F+F(F+*&- F.6#,$*&\"\"%F(F*F(F+F(F*F(F(F3F(F(F*!\"$F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Since the periodic function " } {XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 40 " has period 4, th e Laplace transform of " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" } {TEXT -1 42 " is obtained by dividing the transform of " }{XPPEDIT 18 0 "`f*`(t)" "6#-%#f*G6#%\"tG" }{TEXT -1 3 " by" }{XPPEDIT 18 0 "``(1-e xp(-4*s));" "6#-%!G6#,&\"\"\"F'-%$expG6#,$*&\"\"%F'%\"sGF'!\"\"F/" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Fs/(1-exp(-4*s));\n``=simplify(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,$**\"\"#\"\"\",,%\"sG!\"\"F&F&*&F%F&-%$expG6#,$ *&F%F&F(F&F)F&F)*&-F,6#,$*&\"\"%F&F(F&F)F&F(F&F&F1F&F&F(!\"$,&F&F&F1F) F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$**\"\"#\"\"\",*-%$expG6# ,$*&F'F(%\"sGF(!\"\"F(*&F*F(F/F(F(F/F(F(F0F(,&F*F(F(F(F0F/!\"$F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Alternatively, we can start wit h the piecewise definition of " }{XPPEDIT 18 0 "`f*`(t)" "6#-%#f*G6#% \"tG" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f*`(t) = PIECEWISE([2*t-t^2, 0 <= t and t < 2],[t^2-6*t+8, 2 <= t and t < 4],[0, 4 <= t]);" "6#/-%#f*G6#%\"tG-%*PIECEWISEG6%7$,&*&\" \"#\"\"\"F'F/F/*$F'F.!\"\"31\"\"!F'2F'F.7$,(*$F'F.F/*&\"\"'F/F'F/F1\" \")F/31F.F'2F'\"\"%7$F41F?F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 230 "`f*` := t - > piecewise(t<2,2*t-t^2,t<4,t^2-6*t+8,t>=4,0):\n'`f*`(t)'=`f*`(t);\nco nvert(`f*`(t),Heaviside):\n`f*` := unapply(%,t):\n'`f*`(t)'=`f*`(t);\n `Laplace transform`=inttrans[laplace](`f*`(t),t,s);\n``=normal(rhs(%)) ;\nFs := rhs(%):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f*G6#%\"tG-%*P IECEWISEG6%7$,&*&\"\"#\"\"\"F'F/F/*$)F'F.F/!\"\"2F'F.7$,(F0F/*&\"\"'F/ F'F/F2\"\")F/2F'\"\"%7$\"\"!1F:F'" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/ -%#f*G6#%\"tG,2*&\"\"#\"\"\"F'F+F+*$)F'F*F+!\"\"*(F*F+-%*HeavisideG6#, &F*F.F'F+F+F-F+F+*(\"\")F+F0F+F'F+F.*&F5F+F0F+F+*&-F16#,&F'F+\"\"%F.F+ F-F+F.*(\"\"'F+F8F+F'F+F+*&F5F+F8F+F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG,,*&\"\"#\"\"\"%\"sG!\"#F(*&F'F(F)!\"$!\"\"*( \"\"%F(-%$expG6#,$*&F'F(F)F(F-F(F)F,F(*(F'F(-F16#,$*&F/F(F)F(F-F(F)F*F -*(F'F(F6F(F)F,F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*(\"\"#\"\" \",,%\"sG!\"\"F(F(*&F'F(-%$expG6#,$*&F'F(F*F(F+F(F+*&-F.6#,$*&\"\"%F(F *F(F+F(F*F(F(F3F(F(F*!\"$F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 " As before the Laplace transform of " } {XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 42 " is obtained by d ividing the transform of " }{XPPEDIT 18 0 "`f*`(t)" "6#-%#f*G6#%\"tG" }{TEXT -1 3 " by" }{XPPEDIT 18 0 "``(1-exp(-4*s));" "6#-%!G6#,&\"\"\"F '-%$expG6#,$*&\"\"%F'%\"sGF'!\"\"F/" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Fs/(1-exp (-4*s));\n``=simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"\"# \"\"\",,%\"sG!\"\"F&F&*&F%F&-%$expG6#,$*&F%F&F(F&F)F&F)*&-F,6#,$*&\"\" %F&F(F&F)F&F(F&F&F1F&F&F(!\"$,&F&F&F1F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$**\"\"#\"\"\",*-%$expG6#,$*&F'F(%\"sGF(!\"\"F(*&F *F(F/F(F(F/F(F(F0F(,&F*F(F(F(F0F/!\"$F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "We can also obtain this result by using the integral form ula:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[f(t)];" "6 #*&%\"LG\"\"\"7#-%\"fG6#%\"tGF%" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "1/( 1-exp(-p*s));" "6#*&\"\"\"F$,&F$F$-%$expG6#,$*&%\"pGF$%\"sGF$!\"\"F-F- " }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(t)*exp(-t*s),t = 0 .. p);" "6# -%$IntG6$*&-%\"fG6#%\"tG\"\"\"-%$expG6#,$*&F*F+%\"sGF+!\"\"F+/F*;\"\"! %\"pG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " } {XPPEDIT 18 0 "p = 4;" "6#/%\"pG\"\"%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"t G" }{TEXT -1 16 " coincides with " }{XPPEDIT 18 0 "`f*`(t)" "6#-%#f*G6 #%\"tG" }{TEXT -1 19 " over the interval " }{XPPEDIT 18 0 "[0, 4];" "6 #7$\"\"!\"\"%" }{TEXT -1 5 ", so " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#% \"tG" }{TEXT -1 37 " in the integrand can be replaced by " }{XPPEDIT 18 0 "`f*`(t)" "6#-%#f*G6#%\"tG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f*`( t) = 2*t-t^2+u[2](t)*(2*t^2-8*t+8)-u[4](t)*(t^2-6*t+8);" "6#/-%#f*G6#% \"tG,**&\"\"#\"\"\"F'F+F+*$F'F*!\"\"*&-&%\"uG6#F*6#F'F+,(*&F*F+*$F'F*F +F+*&\"\")F+F'F+F-F8F+F+F+*&-&F16#\"\"%6#F'F+,(*$F'F*F+*&\"\"'F+F'F+F- F8F+F+F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = PIECEWISE([2*t-t^2, 0 <= \+ t and t < 2],[t^2-6*t+8, 2 <= t and t < 4],[0, 4 <= t]);" "6#/%!G-%*PI ECEWISEG6%7$,&*&\"\"#\"\"\"%\"tGF,F,*$F-F+!\"\"31\"\"!F-2F-F+7$,(*$F-F +F,*&\"\"'F,F-F,F/\"\")F,31F+F-2F-\"\"%7$F21F=F-" }{TEXT -1 2 " ." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "`f*` := t -> piecewise(t<2,2*t-t^2,t<4,t^2-6*t+8,t>=4,0):\n'`f*`( t)'=`f*`(t);\nInt('`f*`(t)'*exp(-s*t),t=0..4)/(1-exp(-4*s));\n``=value (%);\n``=simplify(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f*G6 #%\"tG-%*PIECEWISEG6%7$,&*&\"\"#\"\"\"F'F/F/*$)F'F.F/!\"\"2F'F.7$,(F0F /*&\"\"'F/F'F/F2\"\")F/2F'\"\"%7$\"\"!1F:F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$IntG6$*&-%#f*G6#%\"tG\"\"\"-%$expG6#,$*&%\"sGF,F+F ,!\"\"F,/F+;\"\"!\"\"%*$,&F,F,-F.6#,$*&F7F,F2F,F3F3F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$**\"\"#\"\"\",,%\"sGF(F(!\"\"*&-%$expG6#,$*& \"\"%F(F*F(F+F(F*F(F+F-F+*&F'F(-F.6#,$*&F'F(F*F(F+F(F(F(F*!\"$,&F(F(F- F+F+F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$**\"\"#\"\"\",*-%$expG 6#,$*&F'F(%\"sGF(!\"\"F(*&F*F(F/F(F(F/F(F(F0F(,&F*F(F(F(F0F/!\"$F(" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "A procedure " }{TEXT 0 1 "u" }{TEXT -1 41 " which implements the unit step function " } {XPPEDIT 18 0 "u[a](t)" "6#-&%\"uG6#%\"aG6#%\"tG" }{TEXT -1 75 " and a procedure for \ncollecting together terms with the same denominator: \+ " }{TEXT 0 19 "collect_like_denoms" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "u" {MPLTEXT 1 0 198 "u := proc(t)\n local a; \n if type(procname,specindex(algebraic,u)) and type(t,algebraic) th en\n a := op(1,procname); \n piecewise(t=a,1);\n e lse 'procname'(t)\n end if;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "collect_like_denoms" {MPLTEXT 1 0 600 "c ollect_like_denoms := proc(ee::algebraic)\n local i,j,denoms,terms,t erm,dn;\n if op(0,ee)=`+` then\n denoms := \{\};\n for i t o nops(ee) do\n denoms := \{op(denoms),denom(op(i,ee))\};\n \+ end do;\n end if;\n denoms := [op(denoms)];\n terms := [];\n \+ for i to nops(denoms) do\n dn := op(i,denoms);\n term := 0 ;\n for j to nops(ee) do\n if denom(op(j,ee))=dn then\n \+ term := term+op(j,ee);\n end if;\n end do;\n \+ terms := [op(terms),term];\n end do;\n terms := map(normal,te rms);\n add(op(i,terms),i=1..nops(terms)); \nend proc:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }} {PARA 0 "" 0 "" {TEXT -1 28 "Find the Laplace transforms " }{XPPEDIT 18 0 "L*[f(t)];" "6#*&%\"LG\"\"\"7#-%\"fG6#%\"tGF%" }{TEXT -1 37 " of \+ the following periodic functions " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#% \"tG" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(t) = PIECEWISE([1, 0 <= t and t < 1],[0, 1 \+ <= t and t < 2]);" "6#/-%\"fG6#%\"tG-%*PIECEWISEG6$7$\"\"\"31\"\"!F'2F 'F,7$F/31F,F'2F'\"\"#" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 5 " and " }{XPPEDIT 18 0 "f(t);" "6#-% \"fG6#%\"tG" }{TEXT -1 28 " is periodic with period 2. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "plot(( 1+(-1)^floor(t))/2,t=0..7.5,-0.2..1.2,thickness=2,\n color=C OLOR(RGB,0,.5,.2),labels=[t,`f(t)`],ytickmarks=3);" }}{PARA 13 "" 1 " " {GLPLOT2D 668 166 166 {PLOTDATA 2 "6(-%'CURVESG6#7jr7$$\"\"!F)$\"\" \"F)7$$\"3$*****\\ilyM;!#=F*7$$\"3'***\\P4c?dIF/F*7$$\"3s***\\Pfkol%F/ F*7$$\"3%3+]i:FrE'F/F*7$$\"3!***\\(=UO(pyF/F*7$$\"3a**\\i!R[Eh)F/F*7$$ \"3=**\\Pf.cb$*F/F*7$$\"3E****\\P4(ya*F/F*7$$\"3M**\\i::=S(*F/F*7$$\"3 $**\\(o/oLO)*F/F*7$$\"3U***\\P4#\\K**F/F*7$$\"3g[7GQ(p0)**F/F*7$$\"3*) \\7GQZ'G+\"!#FTF(7$$\"3#)**** \\is#p&>FTF(7$$\"3#**\\(=#*3Bx>FTF(7$$\"3!)**\\(=_Mv*>FTF(7$$\"3suoHH/ h-?FTF*7$$\"3&)\\(=nL'o2?FTF*7$$\"3bC19WAw7?FTF*7$$\"3o*\\i::Qy,#FTF*7 $$\"3'*\\iSm**)z-#FTF*7$$\"3y***\\7yT\"Q?FTF*7$$\"3$****\\(=qS;@FTF*7$ $\"31++DcAn%>#FTF*7$$\"3-+vVBp\"oL#FTF*7$$\"3/++D\"oSe]#FTF*7$$\"3#)** ***\\(f-\\EFTF*7$$\"3t*\\P4^%e:GFTF*7$$\"3)*\\(=Ux+$*)GFTF*7$$\"3!)*** *\\Pq,jHFTF*7$$\"3WVt_So7tHFTF*7$$\"3a(oaNkOK)HFTF*7$$\"3gf$o]a\"H))HF TF*7$$\"3?J?eYkM$*HFTF*7$$\"3!Gq&4[8S)*HFTF*7$$\"3&[P4'\\iX.IFTF(7$$\" 3;iSmbenBIFTF(7$$\"3#*\\(=&RMFTF(7 $$\"3O+vo/h5(e$FTF(7$$\"3;+]7y!)HYPFTF(7$$\"3$)\\7GQf(*GQFTF(7$$\"3%** \\P%)z`;\"RFTF(7$$\"3?]Pfe%Rw%RFTF(7$$\"3/++v=^i$)RFTF(7$$\"3EJ&piKB\" ))RFTF(7$$\"3[i!*yL:i#*RFTF(7$$\"3q$f38u>r*RFTF(7$$\"3%\\7G)[zh,SFTF*7 $$\"3R(=nQO91,%FTF*7$$\"3&)\\i!*y2h>SFTF*7$$\"3wuV)*3OgPSFTF*7$$\"3o* \\i!RkfbSFTF*7$$\"3e\\(=nVFL8%FTF*7$$\"3Q+]PM%e5@%FTF*7$$\"3%)****\\() emrVFTF*7$$\"3A+](o%*)yGXFTF*7$$\"3'**\\Pfe83o%FTF*7$$\"3?]7.2B@lZFTF* 7$$\"3c**\\7G5h\\[FTF*7$$\"3)*\\(o/.Hv)[FTF*7$$\"3^*\\7G.Za#\\FTF*7$$ \"3GuV)R.1W%\\FTF*7$$\"31\\i:N]Oj\\FTF*7$$\"3)o=Ud`WG(\\FTF*7$$\"3sC\" Gj.CB)\\FTF*7$$\"3i$4@myjq)\\FTF*7$$\"3aiS\"p`.=*\\FTF*7$$\"3cIq?(GVl* \\FTF*7$$\"3Z****\\PIG,]FTF(7$$\"3C**\\PfTD#3&FTF(7$$\"3!****\\7GDK;&F TF(7$$\"3m*\\PM-p*4`FTF(7$$\"3)3+]Pp)RqaFTF(7$$\"3$**\\P4^]8i&FTF(7$$ \"3K+voa'Q\"zdFTF(7$$\"3))\\(o/.si&eFTF(7$$\"3Y***\\iS0M$fFTF(7$$\"3/P MF!G$f`fFTF(7$$\"3iuoHa6ytfFTF(7$$\"3'Qf384vQ)fFTF(7$$\"3?7.KG!pR*fFTF (7$$\"3#= b\"pFTF*7$$\"35+DcE\"oE&pFTF*7$$\"3_u=<^DCrpFTF*7$$\"3%)\\7yvp\")*)pFT F*7$$\"3)HfL>egW*pFTF*7$$\"3/Pf3)=/\"**pFTF*7$$\"34\"GQUzZP+(FTF(7$$\" 39D1R+9R3qFTF(7$$\"3N7`p7'yw,(FTF(7$$\"3c*****\\#e'p-(FTF(7$$\"34](ozm #=1rFTF(7$$\"3i+v$4^*R&=(FTF(7$$\"3Q+D1k2/PtFTF(7$$\"3++++++++vFTF(-%& COLORG6&%$RGBGF($\"\"&!\"\"$\"\"#Fecl-%*THICKNESSG6#Fgcl-%+AXESLABELSG 6$%\"tG%%f(t)G-%*AXESTICKSG6$%(DEFAULTG\"\"$-%%VIEWG6$;F($\"#vFecl;$! \"#Fecl$\"#7Fecl" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 43.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "An s " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 189 "`f*` := t -> 1 - Heaviside(t-1):\n'`f*`(t)'=`f*`(t); \n``=convert(`f*`(t),piecewise);\nplot(`f*`(t),t=0..4.5,-0.2..1.2,thic kness=2,\n color=COLOR(RGB,0,.5,.2),labels=[t,`f*(t)`],ytickmarks=3) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f*G6#%\"tG,&\"\"\"F)-%*Heavis ideG6#,&F)!\"\"F'F)F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%*PIECEW ISEG6%7$\"\"\"2%\"tGF)7$%*undefinedG/F+F)7$\"\"!2F)F+" }}{PARA 13 "" 1 "" {GLPLOT2D 496 173 173 {PLOTDATA 2 "6(-%'CURVESG6#7hn7$$\"\"!F)$\" \"\"F)7$$\"3e*****\\P>(3)*!#>F*7$$\"3?+]ilLKM=!#=F*7$$\"3E++Dc(=Tz#F3F *7$$\"31++v$Hw-w$F3F*7$$\"3\\**\\7`=%=s%F3F*7$$\"3'***\\i:iL8cF3F*7$$ \"3V**\\i!*pUOlF3F*7$$\"35+]i!z)3\"\\(F3F*7$$\"3>**\\7y*)oU%)F3F*7$$\" 3;+D\"y&y5K*)F3F*7$$\"39,+]Pn_@%*F3F*7$$\"3>+DJXp1P'*F3F*7$$\"3O+]7`rg _)*F3F*7$$\"3nD\"y]?#\\1**F3F*7$$\"3))\\7.dsPg**F3F*7$$\"358y+$y>t)**F 3F*7$$\"3jP%)*3BE9+\"!#F[oF(7$$\"3O+]il$[nMF[oF(7$$\"3/++vVK/gNF[oF(7$$\"3!)*\\i!R]% pl$F[oF(7$$\"3]+++&)HF]PF[oF(7$$\"3/+]P*G9d%QF[oF(7$$\"3E+Dc\"Hl.%RF[o F(7$$\"3x****\\K(Rt-%F[oF(7$$\"3p**\\(oDAq7%F[oF(7$$\"3W+++&\\zh@%F[oF (7$$\"3m*\\ilqR7J%F[oF(7$$\"3))*\\P%eWA-WF[oF(7$$\"3++++++++XF[oF(-%+A XESLABELSG6$%\"tG%&f*(t)G-%&COLORG6&%$RGBGF($\"\"&!\"\"$\"\"#Few-%*THI CKNESSG6#Fgw-%*AXESTICKSG6$%(DEFAULTG\"\"$-%%VIEWG6$;F($\"#XFew;$!\"#F ew$\"#7Few" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "C urve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "`f*` := t -> 1 - Heaviside(t-1):\n'`f*`(t)'=`f*`(t); \n`Laplace transform`=inttrans[laplace](rhs(%),t,s)/(1-exp(-2*s));\n`` =combine(simplify(rhs(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f*G 6#%\"tG,&\"\"\"F)-%*HeavisideG6#,&F)!\"\"F'F)F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG*&,&*&\"\"\"F(%\"sG!\"\"F(*&-%$exp G6#,$F)F*F(F)F*F*F(,&F(F(-F-6#,$*&\"\"#F(F)F(F*F*F*" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%!G*&\"\"\"F&*&,&F&F&-%$expG6#,$%\"sG!\"\"F&F&F-F&F. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "L*[f(t )]=1/((1+exp(-s))*s)" "6#/*&%\"LG\"\"\"7#-%\"fG6#%\"tGF&*&F&F&*&,&F&F& -%$expG6#,$%\"sG!\"\"F&F&F3F&F4" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 29 "_____________________________" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 29 "_____________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(t) = PIECEWISE([t, \+ 0 <= t and t < 1],[1, 1 <= t and t < 2]);" "6#/-%\"fG6#%\"tG-%*PIECEWI SEG6$7$F'31\"\"!F'2F'\"\"\"7$F031F0F'2F'\"\"#" }{TEXT -1 3 " , " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 " and " } {XPPEDIT 18 0 "f(t);" "6#-%\"fG6#%\"tG" }{TEXT -1 28 " is periodic wit h period 2. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "g := t -> piecewise(t<1,t,1):\nf := x -> g(x-2*fl oor(x/2)):\nplot(f(t),t=0..7.5,0..1.2,thickness=2,\n color=COLOR (RGB,.8,.5,0),labels=[t,`f(t)`],ytickmarks=3);" }}{PARA 13 "" 1 "" {GLPLOT2D 588 204 204 {PLOTDATA 2 "6(-%'CURVESG6#7_r7$$\"\"!F)F(7$$\"3 m****\\7G$R<)!#>F+7$$\"3$*****\\ilyM;!#=F/7$$\"35+v$f3'*fM#F1F37$$\"3' ***\\P4c?dIF1F67$$\"3%)*\\i:5Nq&QF1F97$$\"3s***\\Pfkol%F1F<7$$\"3%3++] (e*>Y&F1F?7$$\"3%3+]i:FrE'F1FB7$$\"3O+D1*yJ%oqF1FE7$$\"3!***\\(=UO(pyF 1FH7$$\"3a**\\i!R[Eh)F1FK7$$\"3=**\\Pf.cb$*F1FN7$$\"3M**\\i::=S(*F1FQ7 $$\"3&**\\(=n-[75!#<$\"\"\"F)7$$\"3'**\\7GQU40\"FVFW7$$\"3)**\\P%)\\/% *3\"FVFW7$$\"3++vV['f*o6FVFW7$$\"3-+vV)z9&[7FVFW7$$\"3,+voH[629FVFW7$$ \"3/++DcWDq:FVFW7$$\"32+]7Gz%Rr\"FVFW7$$\"3=+v$fLI[z\"FVFW7$$\"33++vVF rv=FVFW7$$\"31+]7.+K;>FVFW7$$\"3#)****\\is#p&>FVFW7$$\"3#**\\(=#*3Bx>F VFW7$$\"3!)**\\(=_Mv*>FVFW7$$\"3suoHH/h-?FV$\"3q;Z(oHH/h#!#?7$$\"3&)\\ (=nL'o2?FV$\"3E`)\\(=nL'o(F_q7$$\"3bC19WAw7?FV$\"3eaC19WAw7F-7$$\"3o* \\i::Qy,#FV$\"3Ao*\\i::Qy\"F-7$$\"3'*\\iSm**)z-#FV$\"3a&*\\iSm**)z#F-7 $$\"3y***\\7yT\"Q?FV$\"3Wy***\\7yT\"QF-7$$\"3$****\\(=qS;@FV$\"3C**** \\(=qS;\"F17$$\"31++DcAn%>#FV$\"3k++]iDsY>F17$$\"3/]P%)*eWdE#FV$\"3Y+v V)*eWdEF17$$\"3-+vVBp\"oL#FV$\"3E+]PM#p\"oLF17$$\"3.]PM-)G8U#FV$\"3J+v VB!)G8UF17$$\"3/++D\"oSe]#FV$\"3M++]7oSe]F17$$\"39+]7GLVxDFV$\"3Y,+D\" GLVx&F17$$\"3#)*****\\(f-\\EFV$\"35)*****\\(f-\\'F17$$\"3x\\(oHC0Bt#FV $\"3q(\\(oHC0BtF17$$\"3t*\\P4^%e:GFV$\"3G(*\\P4^%e:)F17$$\"3)*\\(=Ux+$ *)GFV$\"3y*\\(=Ux+$*))F17$$\"3!)****\\Pq,jHFV$\"3%y****\\Pq,j*F17$$\"3 a(oaNkOK)HFV$\"3RvoaNkOK)*F17$$\"3&[P4'\\iX.IFVFW7$$\"3;iSmbenBIFVFW7$ $\"3#*\\(=&RMFVFW7$$\"3O+vo/h5(e$F VFW7$$\"3;+]7y!)HYPFVFW7$$\"3$)\\7GQf(*GQFVFW7$$\"3%**\\P%)z`;\"RFVFW7 $$\"3?]Pfe%Rw%RFVFW7$$\"3/++v=^i$)RFVFW7$$\"3EJ&piKB\"))RFVFW7$$\"3[i! *yL:i#*RFVFW7$$\"3q$f38u>r*RFVFW7$$\"3%\\7G)[zh,SFV$\"3oP\\7G)[zh\"F_q 7$$\"3R(=nQO91,%FV$\"3IR(=nQO91\"F-7$$\"3&)\\i!*y2h>SFV$\"3%[)\\i!*y2h >F-7$$\"3wuV)*3OgPSFV$\"3!fZP%)*3OgPF-7$$\"3o*\\i!RkfbSFV$\"3)p'*\\i!R kfbF-7$$\"3e\\(=nVFL8%FV$\"3\"e\\(=nVFL8F17$$\"3Q+]PM%e5@%FV$\"3\"Q+]P M%e5@F17$$\"37+v$4;i8H%FV$\"39,]P4;i8HF17$$\"3%)****\\()emrVFV$\"3[)** **\\()emr$F17$$\"3[+v= " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 210 "`f*` := t -> t + Heaviside( t-1)*(1-t)-Heaviside(t-2):\n'`f*`(t)'=`f*`(t);\n``=convert(`f*`(t),pie cewise);\nplot(`f*`(t),t=0..6.5,-0.2..1.2,thickness=2,\n color=COLOR (RGB,.8,.5,0),labels=[t,`f*(t)`],ytickmarks=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f*G6#%\"tG,(F'\"\"\"*&-%*HeavisideG6#,&F)!\"\"F'F)F ),&F)F)F'F/F)F)-F,6#,&\"\"#F/F'F)F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%!G-%*PIECEWISEG6'7$%\"tG2F)\"\"\"7$%*undefinedG/F)F+7$F+2F)\"\"#7$F -/F)F17$\"\"!2F1F)" }}{PARA 13 "" 1 "" {GLPLOT2D 476 191 191 {PLOTDATA 2 "6(-%'CURVESG6#7co7$$\"\"!F)F(7$$\"3Omm;/^2%3(!#>F+7$$\"3F LL$3-:oT\"!#=F/7$$\"3n*\\7yg'>L?F1F37$$\"3Mm;z%>y&\\EF1F67$$\"3)**\\(o aPwULF1F97$$\"31LLe9$\\f.%F1F<7$$\"3wLLLe(HPt%F1F?7$$\"3\"RL$3--^JaF1F B7$$\"3z*\\(=O)485!#<$\"\"\"F)7$$\"3s*\\7.Esv/\"FfnFgn7$$ \"3?Lekeh/#3\"FfnFgn7$$\"3[mT&)QGx]6FfnFgn7$$\"3)**\\i!>&*\\>7FfnFgn7$ $\"3emmT&>()3O\"FfnFgn7$$\"3IL$3x`@a[\"FfnFgn7$$\"35++D6xhD;FfnFgn7$$ \"3!****\\Pa*QmU$=FfnFgn7$$\"3,++v)G\\?!>FfnFgn 7$$\"3KeR(**HZG$>FfnFgn7$$\"3U;z>6`kj>FfnFgn7$$\"3o&*)4oJW!z>FfnFgn7$$ \"3su=UALW%*>FfnFgn7$$\"3kp[#Q2$H)*>FfnFgn7$$\"3NkyADG9-?FfnF(7$$\"31f 3jwD*f+#FfnF(7$$\"3w`Q.GB%)4?FfnF(7$$\"3iV)R3$=ag%FfnF(7$$\"3B++D,A,TZFfnF(7$$\"3[l\"z%4r$=([Ffn F(7$$\"3;+D1Moe3]FfnF(7$$\"3slmT&o%GU^FfnF(7$$\"3E*\\7`%RD#G&FfnF(7$$ \"3mLLLy41 " 0 "" {MPLTEXT 1 0 167 "`f*` := t -> t + Heaviside(t-1)*(1-t)-Heaviside(t-2):\n'`f*`(t)'= `f*`(t);\n`Laplace transform`=inttrans[laplace](rhs(%),t,s)/(1-exp(-2* s));\n``=combine(simplify(rhs(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%#f*G6#%\"tG,(F'\"\"\"*&-%*HeavisideG6#,&F)!\"\"F'F)F),&F)F)F'F/F)F )-F,6#,&\"\"#F/F'F)F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~tr ansformG*&,(*&\"\"\"F(*$)%\"sG\"\"#F(!\"\"F(*&-%$expG6#,$F+F-F(F+!\"#F -*&-F06#,$*&F,F(F+F(F-F(F+F-F-F(,&F(F(F5F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*(,(\"\"\"!\"\"-%$expG6#,$%\"sGF(F'*&-F*6#,$*&\"\"# F'F-F'F(F'F-F'F'F'F-!\"#,&F'F(F/F'F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "L*[f(t)] = (1-exp(-s)-exp(-2*s)*s)/(s ^2*(1-exp(-2*s)));" "6#/*&%\"LG\"\"\"7#-%\"fG6#%\"tGF&*&,(F&F&-%$expG6 #,$%\"sG!\"\"F3*&-F/6#,$*&\"\"#F&F2F&F3F&F2F&F3F&*&F2F9,&F&F&-F/6#,$*& F9F&F2F&F3F3F&F3" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 29 "_____________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 29 "____________ _________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q3 " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(t) = PIECEWISE([1-t, 0 <= t and t < 1],[0, 1 \+ <= t and t < 2]);" "6#/-%\"fG6#%\"tG-%*PIECEWISEG6$7$,&\"\"\"F-F'!\"\" 31\"\"!F'2F'F-7$F131F-F'2F'\"\"#" }{TEXT -1 3 " , " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 " and " }{XPPEDIT 18 0 "f (t);" "6#-%\"fG6#%\"tG" }{TEXT -1 28 " is periodic with period 2. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "g := t -> piecewise(t<1,1-t,0):\nf := x -> g(x-2*floor(x/2)):\npl ot(f(t),t=0..7.5,0..1.2,thickness=2,\n color=COLOR(RGB,.8,0,.5), labels=[t,`f(t)`],ytickmarks=3);" }}{PARA 13 "" 1 "" {GLPLOT2D 664 164 164 {PLOTDATA 2 "6(-%'CURVESG6#7_r7$$\"\"!F)$\"\"\"F)7$$\"3m****\\ 7G$R<)!#>$\"3Y++v=ng#=*!#=7$$\"3$*****\\ilyM;F2$\"3!)****\\PM@l$)F27$$ \"35+v$f3'*fM#F2$\"3O*\\iS\"R+awF27$$\"3'***\\P4c?dIF2$\"3/+]i!R%zUpF2 7$$\"3%)*\\i:5Nq&QF2$\"3g*\\P%)*['H9'F27$$\"3s***\\Pfkol%F2$\"3G++D1a8 V`F27$$\"3%3++](e*>Y&F2$\"3<*****\\7/!QXF27$$\"3%3+]i:FrE'F2$\"3<***\\ P%G(Gt$F27$$\"3O+D1*yJ%oqF2$\"3k*\\P4@o:$HF27$$\"3!***\\(=UO(pyF2$\"36 +]7yNEI@F27$$\"3a**\\i!R[Eh)F2$\"3Y+]P4;N(Q\"F27$$\"3=**\\Pf.cb$*F2$\" 353+D1kRWkF/7$$\"3M**\\i::=S(*F2$\"3e1+vV[=)f#F/7$$\"3&**\\(=n-[75!#F\\pF(7$$\"3#)****\\is#p&>F\\pF(7$$\"3#**\\(=#*3Bx>F\\pF(7 $$\"3!)**\\(=_Mv*>F\\pF(7$$\"3suoHH/h-?F\\p$\"3%GDJqq&*Q(**F27$$\"3&) \\(=nL'o2?F\\p$\"3Z,D\"GjOJ#**F27$$\"3bC19WAw7?F\\p$\"3aaPfevPs)*F27$$ \"3o*\\i::Qy,#F\\p$\"3=.]P%[=;#)*F27$$\"3'*\\iSm**)z-#F\\p$\"3X+v$fL+, s*F27$$\"3y***\\7yT\"Q?F\\p$\"3;-+](=#e='*F27$$\"3$****\\(=qS;@F\\p$\" 3w++]7)Hf$))F27$$\"31++DcAn%>#F\\p$\"3O****\\PuF`!)F27$$\"3/]P%)*eWdE# F\\p$\"3b*\\i:5aDM(F27$$\"3-+vVBp\"oL#F\\p$\"3u**\\il2$=j'F27$$\"3.]PM -)G8U#F\\p$\"3q*\\il(>r'y&F27$$\"3/++D\"oSe]#F\\p$\"3l****\\(=$fT\\F27 $$\"39+]7GLVxDF\\p$\"3b)**\\(=nmDUF27$$\"3#)*****\\(f-\\EF\\p$\"3*=+++ DS(4NF27$$\"3x\\(oHC0Bt#F\\p$\"3I-DJqv%pn#F27$$\"3t*\\P4^%e:GF\\p$\"3q -]i!*[:W=F27$$\"3)*\\(=Ux+$*)GF\\p$\"3@+D\"yD#*p5\"F27$$\"3!)****\\Pq, jHF\\p$\"3g@++]iH)p$F/7$$\"3a(oaNkOK)HF\\p$\"31Y7`WcLw;F/7$$\"3&[P4'\\ iX.IF\\pF(7$$\"3;iSmbenBIF\\pF(7$$\"3#*\\(=&RMF\\pF(7$$\"3O+vo/h5(e$F\\pF(7$$\"3;+]7y!)HYPF\\p F(7$$\"3$)\\7GQf(*GQF\\pF(7$$\"3%**\\P%)z`;\"RF\\pF(7$$\"3?]Pfe%Rw%RF \\pF(7$$\"3/++v=^i$)RF\\pF(7$$\"3EJ&piKB\"))RF\\pF(7$$\"3[i!*yL:i#*RF \\pF(7$$\"3q$f38u>r*RF\\pF(7$$\"3%\\7G)[zh,SF\\p$\"3i](=<^?Q)**F27$$\" 3R(=nQO91,%F\\p$\"31E\"G8OcQ*)*F27$$\"3&)\\i!*y2h>SF\\p$\"3_,v$4@#*Q!) *F27$$\"3wuV)*3OgPSF\\p$\"3T_i:5R'Ri*F27$$\"3o*\\i!RkfbSF\\p$\"3J.]P4c .W%*F27$$\"3e\\(=nVFL8%F\\p$\"3?/D\"GjDnm)F27$$\"3Q+]PM%e5@%F\\p$\"3=' **\\il:%*)yF27$$\"37+v$4;i8H%F\\p$\"3%))*\\i!Ryj3(F27$$\"3%)****\\()em rVF\\p$\"3_,++D6M$G'F27$$\"3[+v=&RF27$$ \"3'**\\Pfe83o%F\\p$\"3M+]iST'=>$F27$$\"3?]7.2B@lZF\\p$\"3'z\\(oHp(yM# F27$$\"3c**\\7G5h\\[F\\p$\"3V/+v=(*)Q]\"F27$$\"3^*\\7G.Za#\\F\\p$\"3_[ +v=nHbuF/7$$\"3Z****\\PIG,]F\\pF(7$$\"3C**\\PfTD#3&F\\pF(7$$\"3!****\\ 7GDK;&F\\pF(7$$\"3m*\\PM-p*4`F\\pF(7$$\"3)3+]Pp)RqaF\\pF(7$$\"3$**\\P4 ^]8i&F\\pF(7$$\"3K+voa'Q\"zdF\\pF(7$$\"3))\\(o/.si&eF\\pF(7$$\"3Y***\\ iS0M$fF\\pF(7$$\"3/PMF!G$f`fF\\pF(7$$\"3iuoHa6ytfF\\pF(7$$\"3'Qf384vQ) fF\\pF(7$$\"3?7.KG!pR*fF\\pF(7$$\"3#= " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 195 "`f*` := t -> 1-t-Heaviside(t-1)*(1-t):\n'`f*`(t)'=`f *`(t);\n``=convert(`f*`(t),piecewise);\nplot(`f*`(t),t=0..4.5,-0.2..1. 2,thickness=2,\n color=COLOR(RGB,.8,0,.5),labels=[t,`f*(t)`],ytickma rks=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f*G6#%\"tG,(\"\"\"F)F'! \"\"*&-%*HeavisideG6#,&F)F*F'F)F),&F)F)F'F*F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%*PIECEWISEG6%7$,&\"\"\"F*%\"tG!\"\"2F+F*7$%*undef inedG/F+F*7$\"\"!2F*F+" }}{PARA 13 "" 1 "" {GLPLOT2D 460 182 182 {PLOTDATA 2 "6(-%'CURVESG6#7in7$$\"\"!F)$\"\"\"F)7$$\"3z****\\(ofV!\\! #>$\"3g***\\7.k&4&*!#=7$$\"3e*****\\P>(3)*F/$\"3K++]i!G\">!*F27$$\"3%* *\\i:l(f29F2$\"3/+vV[BS#f)F27$$\"3?+]ilLKM=F2$\"3!)**\\PMmnl\")F27$$\" 3E++Dc(=Tz#F2$\"3s***\\PC\")e?(F27$$\"31++v$Hw-w$F2$\"3&****\\iqB(RiF2 7$$\"3\\**\\7`=%=s%F2$\"3]+](o9e\"y_F27$$\"3'***\\i:iL8cF2$\"3/+]P%yjm Q%F27$$\"3V**\\i!*pUOlF2$\"3c+]P4IdjMF27$$\"35+]i!z)3\"\\(F2$\"3!***\\ P47\"*3DF27$$\"3k**\\P%))))o'zF2$\"3O+]i:66L?F27$$\"3>**\\7y*)oU%)F2$ \"3#3+v=-6tb\"F27$$\"3;+D\"y&y5K*)F2$\"3%)*\\(=U@*y1\"F27$$\"39,+]Pn_@ %*F2$\"3k))***\\iKZy&F/7$$\"3>+DJXp1P'*F2$\"37)*\\(oaI$HOF/7$$\"3O+]7` rg_)*F2$\"3_'**\\(o%GRZ\"F/7$$\"3nD\"y]?#\\1**F2$\"3ALu=#\\z2N*!#?7$$ \"3))\\7.dsPg**F2$\"3W7](oHuA'RF]q7$$\"3jP%)*3BE9+\"!#FfqF(7$$\"3O+]il$[nMFfqF(7$$\"3/++vVK/gNFfqF(7$$\"3!) *\\i!R]%pl$FfqF(7$$\"3]+++&)HF]PFfqF(7$$\"3/+]P*G9d%QFfqF(7$$\"3E+Dc\" Hl.%RFfqF(7$$\"3x****\\K(Rt-%FfqF(7$$\"3p**\\(oDAq7%FfqF(7$$\"3W+++&\\ zh@%FfqF(7$$\"3m*\\ilqR7J%FfqF(7$$\"3))*\\P%eWA-WFfqF(7$$\"3++++++++XF fqF(-%+AXESLABELSG6$%\"tG%&f*(t)G-%&COLORG6&%$RGBG$\"\")!\"\"F($\"\"&F ]z-%*THICKNESSG6#\"\"#-%*AXESTICKSG6$%(DEFAULTG\"\"$-%%VIEWG6$;F($\"#X F]z;$!\"#F]z$\"#7F]z" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "`f*` := t -> 1-t-Heaviside(t-1)*(1 -t):\n'`f*`(t)'=`f*`(t);\n`Laplace transform`=inttrans[laplace](rhs(%) ,t,s)/(1-exp(-2*s));\n``=combine(simplify(rhs(%)));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%#f*G6#%\"tG,(\"\"\"F)F'!\"\"*&-%*HeavisideG6#,&F)F *F'F)F),&F)F)F'F*F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~tr ansformG*&,(*&\"\"\"F(%\"sG!\"\"F(*&F(F(*$)F)\"\"#F(F*F**&-%$expG6#,$F )F*F(F)!\"#F(F(,&F(F(-F16#,$*&F.F(F)F(F*F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*(,(%\"sG\"\"\"F)!\"\"-%$expG6#,$F(F*F)F)F(!\"#,& F)F*-F,6#,$*&\"\"#F)F(F)F*F)F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "L*[f(t)] = (s-1+exp(-s))/(s^2*(1-exp(- 2*s)));" "6#/*&%\"LG\"\"\"7#-%\"fG6#%\"tGF&*&,(%\"sGF&F&!\"\"-%$expG6# ,$F.F/F&F&*&F.\"\"#,&F&F&-F16#,$*&F5F&F.F&F/F/F&F/" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 29 "_____________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 29 "_____________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q4 " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(t) = PIECEWISE([t, 0 <= t and t < 1],[1, 1 <= t and t < 2],[3-t, 2 <= t and t < 3]);" "6#/-%\"fG6#%\"tG-%*PIECEWISE G6%7$F'31\"\"!F'2F'\"\"\"7$F031F0F'2F'\"\"#7$,&\"\"$F0F'!\"\"31F5F'2F' F8" }{TEXT -1 3 " , " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 " and " }{XPPEDIT 18 0 "f(t);" "6#-%\"fG6#%\"tG" } {TEXT -1 28 " is periodic with period 3. 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GB%)4?Ffn$\"3Oi9m>nd,**F17$$\"3.LekL8CD?Ffn$\"3qp;ajmeZ(*F17$$\"3y\\7. i\\[)4#Ffn$\"35-voz.::!*F17$$\"3ammT!fG<<#Ffn$\"3_ML$e49FG)F17$$\"3fm; /^bxLAFfn$\"39MLe*[WAm(F17$$\"3immm6D#eH#Ffn$\"3zLLL$)[xTqF17$$\"35$eR s(y*zO#Ffn$\"3'*oTgF7-?jF17$$\"3f*\\7GCt,W#Ffn$\"37/](=dn#)f&F17$$\"3' He*)4ngS]#Ffn$\"3SqT5!H$Rf\\F17$$\"3Mmm;*4[zc#Ffn$\"3pOLL3!>0K%F17$$\" 3H$e*[LF/QEFfn$\"33nT5lEd>OF17$$\"3\")*\\7yOP\"3FFfn$\"3$>+v=KE'=HF17$ $\"3'***\\iNS)[x#Ffn$\"3K++vV'f6D#F17$$\"37+vV.2jTGFfn$\"3s)*\\ilHp$e \"F17$$\"3`$e9\"\\QF6HFfn$\"3gY;a)3:E())F-7$$\"3]m;z%*p\"4)HFfn$\"3L]L $3_+$3>F-7$$\"3OgxGj7\"*))HFfn$\"3TkRArO()36F-7$$\"3AaQyJb!p*HFfn$\"3% [yXh@oW4$!#?7$$\"32[*z-!)**[+$FfnF(7$$\"3\\TgxoS*G,$FfnF(7$$\"3wG#odg# ))GIFfnF(7$$\"3[;/wU6([/$FfnF(7$$\"3[\"zWn@[o2$FfnF(7$$\"3Zm\"H2HD)3JF fnF(7$$\"3w\\(=#z%3y<$FfnF(7$$\"31L$3xm\"zYKFfnF(7$$\"37Leke**4!R$FfnF (7$$\"3$)*\\(=Z-&[^$FfnF(7$$\"35L$ek(Re\\OFfnF(7$$\"3m****\\-rx)y$FfnF (7$$\"3m**\\i?/&\\#RFfnF(7$$\"3Q*\\7y50n0%FfnF(7$$\"3?**\\PCi*H?%FfnF( 7$$\"3@mm;*HXWL%FfnF(7$$\"3/++vV_zuWFfnF(7$$\"3/Lek`J(>g%FfnF(7$$\"3B+ +D,A,TZFfnF(7$$\"3[l\"z%4r$=([FfnF(7$$\"3;+D1Moe3]FfnF(7$$\"3slmT&o%GU ^FfnF(7$$\"3E*\\7`%RD#G&FfnF(7$$\"3mLLLy41 " 0 "" {MPLTEXT 1 0 222 "`f*` := t -> t+Heaviside(t- 1)*(1-t)+\n Heaviside(t-2)*(2-t)+Heaviside(t-3)*(t-3):\n'`f*` (t)'=`f*`(t);\n`Laplace transform`=inttrans[laplace](rhs(%),t,s)/(1-ex p(-3*s));\n``=normal(rhs(%));\n``=combine(simplify(rhs(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f*G6#%\"tG,*F'\"\"\"*&-%*HeavisideG6#,&F )!\"\"F'F)F),&F)F)F'F/F)F)*&-F,6#,&\"\"#F/F'F)F),&F5F)F'F/F)F)*&-F,6#, &F'F)\"\"$F/F)F:F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~tra nsformG*&,**&\"\"\"F(*$)%\"sG\"\"#F(!\"\"F(*&-%$expG6#,$F+F-F(F+!\"#F- *&-F06#,$*&F,F(F+F(F-F(F+F3F-*&-F06#,$*&\"\"$F(F+F(F-F(F+F3F(F(,&F(F(F :F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*(,*\"\"\"!\"\"-%$expG6#, $%\"sGF(F'-F*6#,$*&\"\"#F'F-F'F(F'-F*6#,$*&\"\"$F'F-F'F(F(F'F-!\"#,&F' F(F3F'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*(,&\"\"\"F'-%$expG6#, $*&\"\"#F'%\"sGF'!\"\"F/F',(F(F'-F)6#,$F.F/F'F'F'F/F.!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "L*[f(t)] = (1-exp(-s )-exp(-2*s)+exp(-3*s))/(s^2*(1-exp(-3*s))" "6#/*&%\"LG\"\"\"7#-%\"fG6# %\"tGF&*&,*F&F&-%$expG6#,$%\"sG!\"\"F3-F/6#,$*&\"\"#F&F2F&F3F3-F/6#,$* &\"\"$F&F2F&F3F&F&*&F2F8,&F&F&-F/6#,$*&F=F&F2F&F3F3F&F3" }{XPPEDIT 18 0 "``= (1-exp(-2*s))/(s^2*(exp(-2*s)+exp(-s)+1))" "6#/%!G*&,&\"\"\"F'- %$expG6#,$*&\"\"#F'%\"sGF'!\"\"F/F'*&F.F-,(-F)6#,$*&F-F'F.F'F/F'-F)6#, $F.F/F'F'F'F'F/" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 29 "_____________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 29 "____________ _________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q5 " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(t) = PIECEW ISE([1, 0 <= t and t < 1],[2, 1 <= t and t < 2],[6-2*t, 2 <= t and t < 3]);" "6#/-%\"fG6#%\"tG-%*PIECEWISEG6%7$\"\"\"31\"\"!F'2F'F,7$\"\"#31 F,F'2F'F27$,&\"\"'F,*&F2F,F'F,!\"\"31F2F'2F'\"\"$" }{TEXT -1 3 " , " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 " and " }{XPPEDIT 18 0 "f(t);" "6#-%\"fG6#%\"tG" }{TEXT -1 28 " is periodic wi th period 3. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "g := t -> piecewise(t<1,1,t<2,2,6-2*t):\nf := x -> g(x-3*floor(x/3)):\nplot(f(t),t=0..9.5,0..2.2,thickness=2,\n \+ color=COLOR(RGB,.9,.7,0),labels=[t,`f(t)`],ytickmarks=3);" }}{PARA 13 "" 1 "" {GLPLOT2D 556 198 198 {PLOTDATA 2 "6(-%'CURVESG6#7ft7$$\"\" !F)$\"\"\"F)7$$\"3CLL$ekH22#!#=F*7$$\"3mm;aQ/YsQF/F*7$$\"31LL3_^p)*eF/ F*7$$\"3dLLek5OQzF/F*7$$\"3Q+D\"G$pM`*)F/F*7$$\"33m;/,GLo**F/F*7$$\"3W PM_mYr-5!#<$\"\"#F)7$$\"3F3F%H0'f35FBFC7$$\"34z>ORuZ95FBFC7$$\"3#*\\7y D)e.-\"FBFC7$$\"3e\"z>')f@@.\"FBFC7$$\"3CL$e9P%)Q/\"FBFC7$$\"3c;a8<*4u 1\"FBFC7$$\"3')*\\7GYN44\"FBFC7$$\"3]mm;al)z8\"FBFC7$$\"37L3_Xw.&=\"FB FC7$$\"3YmT50]Z#G\"FBFC7$$\"3y*\\(okB\"*z8FBFC7$$\"3?L3-y?X\"e\"FBFC7$ $\"3')*\\P4XXBy\"FBFC7$$\"3S$3FW@nc)=FBFC7$$\"3umm\"z(*))*))>FBFC7$$\" 3'**\\PM%[**z?FB$\"32+]78.,S=FB7$$\"3=L$e*32+r@FB$\"3iLL3#e)*zl\"FB7$$ \"3'*\\i!RBEAA#FB$\"33+v=Kvab:FB7$$\"3smT&)e[7FB7$$ \"3>+]iq'QtU#FB$\"3i***\\(eEKX6FB7$$\"39++]KXxyCFB$\"3u*****\\$4XU5FB7 $$\"31+]P%R5-`#FB$\"3y)***\\7@z&R*F/7$$\"3+++Dcik\"e#FB$\"39+++v[2n$)F /7$$\"3E+++SU@JEFB$\"3#[******>:dP(F/7$$\"33++vBAy!o#FB$\"3Q)****\\_bV Q'F/7$$\"3!*****\\2-NIFFB$\"3!>+++&e*HR&F/7$$\"3=++D\">=*zFFB$\"3g'*** *\\7$$\"3eAV^]\"\\L(HFB$\"3%Q[Nr*)p,L&F\\u7$$\"3[7y+)>In)HFB$\"3 )[]P%)RgRl#F\\u7$$\"3)pbaI,bC6,+$ FBF*7$$\"3(o/[#pFB7$$\"3N*\\7G#))3 P^FB$\"3J,]PaB#es\"FB7$$\"3!yXhYX=j=&FB$\"3S%3x14jti\"FB7$$\"3E;/^'3[b B&FB$\"3\\n\"zp#Q!*G:FB7$$\"3su$f$=xx%G&FB$\"3e]7GjXWI9FB7$$\"3;L$3-N2 SL&FB$\"3nLLe*H&)>L\"FB7$$\"3[\\7.xj'[Q&FB$\"3.,v$fCn-B\"FB7$$\"3!e;aQ SDdV&FB$\"3Ro;H#>\\&G6FB7$$\"37#3x1V%e'[&FB$\"3vNekQ6$o-\"FB7$$\"3L*** *\\dMWPbFB$\"3M8++]38^#*F/7$$\"3;\\(oH4*>(e&FB$\"3g;]iS\"=gD)F/7$$\"3! 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\"\"F'F)F)*&-F+6#,&\"\"#F.F'F)F),&\"\"%F)*&F3F)F'F)F.F)F)*&-F+6#,&F'F) \"\"$F.F),&*&F3F)F'F)F)\"\"'F.F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /%2Laplace~transformG*&,**&\"\"\"F(%\"sG!\"\"F(*&-%$expG6#,$F)F*F(F)F* F(*(\"\"#F(-F-6#,$*&F1F(F)F(F*F(F)!\"#F**(F1F(-F-6#,$*&\"\"$F(F)F(F*F( F)F6F(F(,&F(F(F8F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*(,*%\"s G\"\"\"*&-%$expG6#,$F(!\"\"F)F(F)F)*&\"\"#F)-F,6#,$*&F1F)F(F)F/F)F/*&F 1F)-F,6#,$*&\"\"$F)F(F)F/F)F)F)F(!\"#,&F)F/F7F)F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[f(t)]=(s+s*exp(-s)-2 *exp(-2*s)+2*exp(-3*s))/(s^2*(1-exp(-3*s)))" "6#/*&%\"LG\"\"\"7#-%\"fG 6#%\"tGF&*&,*%\"sGF&*&F.F&-%$expG6#,$F.!\"\"F&F&*&\"\"#F&-F16#,$*&F6F& F.F&F4F&F4*&F6F&-F16#,$*&\"\"$F&F.F&F4F&F&F&*&F.F6,&F&F&-F16#,$*&F@F&F .F&F4F4F&F4" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 29 " _____________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 29 "____________ _________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q6 " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(t) = PIECEW ISE([sin(t), 0 <= t and t < Pi],[-1, Pi <= t and t < 2*Pi]);" "6#/-%\" fG6#%\"tG-%*PIECEWISEG6$7$-%$sinG6#F'31\"\"!F'2F'%#PiG7$,$\"\"\"!\"\"3 1F3F'2F'*&\"\"#F6F3F6" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 5 " and " }{XPPEDIT 18 0 "f(t);" "6#-% \"fG6#%\"tG" }{TEXT -1 25 " is periodic with period " }{XPPEDIT 18 0 " 2*P" "6#*&\"\"#\"\"\"%\"PGF%" }{TEXT -1 3 "i. 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2 "6(-%'CURVESG6#7]p7$$\"\"!F)F(7$$\"3imm\"H#[ON5!#=$\"3 :6oH'*f^L5F-7$$\"3CLL$ekH22#F-$\"3K!>Y\")zif0#F-7$$\"3#)*\\(=U]frHF-$ \"3Zb>a))Q0GHF-7$$\"3mm;aQ/YsQF-$\"3CQUFw\")RwPF-7$$\"31LL3_^p)*eF-$\" 3i@JbIf_ibF-7$$\"3dLLek5OQzF-$\"3!esbok![IrF-7$$\"33m;/,GLo**F-$\"3qYY DCyb(R)F-7$$\"37L3_Xw.&=\"!#<$\"3#Q^[GOK^E*F-7$$\"3YmT50]Z#G\"FQ$\"3yu YQpaA(e*F-7$$\"3y*\\(okB\"*z8FQ$\"3Hadot!p$=)*F-7$$\"33L3-$H(HI9FQ$\"3 \"34$*=Oi9!**F-7$$\"3QmTN@Ao![\"FQ$\"3e,(o*3ZUf**F-7$$\"3\"**\\(o\\r1J :FQ$\"3Sz\"[/+4@***F-7$$\"3?L3-y?X\"e\"FQ$\"3O)*3?!GK%****F-7$$\"3w*** \\7Uv;j\"FQ$\"31S?3-WZ\")**F-7$$\"3_m\"zWw)*=o\"FQ$\"3Wf(QBqW$Q**F-7$$ \"3JL$3x5A@t\"FQ$\"37S*yx%>:q)*F-7$$\"3')*\\P4XXBy\"FQ$\"3))*pw06oqx*F -7$$\"3S$3FW@nc)=FQ$\"3q.CF@PO3&*F-7$$\"3umm\"z(*))*))>FQ$\"3*4e=+9W#Q \"*F-7$$\"3=L$e*32+r@FQ$\"3/K@A[>?_#)F-7$$\"3E++v3G!fP#FQ$\"3*\\NkjvW. $pF-7$$\"3+++Dcik\"e#FQ$\"3^p'*H?yS6`F-7$$\"3=++D\">=*zFFQ$\"3:acqE'3% QNF-7$$\"3'oTNr9V*pGFQ$\"3g`56%>3Ko#F-7$$\"37L3-.\"o*fHFQ$\"3[*z%4.kF1 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QF(7$$\"3S++vy; " 0 "" {MPLTEXT 1 0 152 "`f*` := t ->sin(t)+Heavisid e(t-Pi)*(-sin(t)-1)+Heaviside(t-2*Pi):\n'`f*`(t)'=`f*`(t);\n`Laplace t ransform`=inttrans[laplace](rhs(%),t,s)/(1-exp(-2*Pi*s));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f*G6#%\"tG,(-%$sinGF&\"\"\"*&-%*HeavisideG6 #,&%#PiG!\"\"F'F+F+,&F)F2F+F2F+F+-F.6#,&F'F+*&\"\"#F+F1F+F2F+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG*&,**&\"\"\"F(,&* $)%\"sG\"\"#F(F(F(F(!\"\"F(*&-%$expG6#,$*&F,F(%#PiGF(F.F(F)F.F(*&F0F(F ,F.F.*&-F16#,$*(F-F(F,F(F5F(F.F(F,F.F(F(,&F(F(F8F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "1/(1-exp(-2*Pi*s))" " 6#*&\"\"\"F$,&F$F$-%$expG6#,$*(\"\"#F$%#PiGF$%\"sGF$!\"\"F.F." } {XPPEDIT 18 0 "``( (1+exp(-Pi*s))/(s^2+1)+(exp(-2*Pi*s)-exp(-Pi*s))/s) " "6#-%!G6#,&*&,&\"\"\"F)-%$expG6#,$*&%#PiGF)%\"sGF)!\"\"F)F),&*$F0\" \"#F)F)F)F1F)*&,&-F+6#,$*(F4F)F/F)F0F)F1F)-F+6#,$*&F/F)F0F)F1F1F)F0F1F )" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 29 "__________ ___________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 29 "_____________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 0 "" 0 "" {TEXT -1 18 "Code for pictures " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 9 "graph of " }{XPPEDIT 18 0 "y=cos^2*t" "6#/%\"yG*&%$cosG\"\"#%\"t G\"\"\"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 327 "p1 := plot(cos(t)^2,t=0..11.0,colo r=red):\nt1 := plots[textplot]([[11.5,-.05,`t`],[-.3,1.2,`f(t)`]],\n \+ font=[HELVETICA,10]):\nplots[display]([p1,t1],xtickmarks=[0=`0`,1. 57=`p/2`,3.14=`p`,\n 4.71=`3p/2`,6.28=`2p`,7.854=`5p/2`,9.425=`3p`,1 1=`7p/2`], ytickmarks=3,labels=[``,``],font=[SYMBOL,9],view=[-.3. .11.5,-.2..1.2]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 58 "e xtending a function on an interval to a periodic function" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1402 "f : = x -> 0.2+0.7*sin(x)+1/10*cos(5*x):\nf_ := x -> f(x-2*floor(x/2)):\nf a := .3: fb := f(2.):\np1 := plot(f(x),x=0..2,color=red):\np2 := plot( f(x+2),x=-2..0,color=brown):\np3 := plot(f(x-2),x=2..4,color=brown):\n p4 := plot(f(x-4),x=4..6,color=brown):\np5 := plot(f(x-6),x=6..8,color =brown):\np6 := plot([[-2,0],[8,0]],color=black):\np7 := plot([[[-2,0] ,[-2,f(0)]],seq([[2*i,0],[2*i,f(2)]],i=0..4)],\n color= black,linestyle=2):\np8 := plot([[[.7,0],[.7,f(.7)]],[[4.7,0],[4.7,f(. 7)]]],\n color=blue,linestyle=3):\np9 := plot([[[-2,fa] ,[2,fa],[4,fa],[6,fa]]$3],style=point,\n symbol=[circle,diamond,cr oss],color=brown):\np10 := plot([[0,fb],[4,fb],[6,fb],[8,fb]],style=po int,symbol=circle,color=brown):\np11 := plot([[[0,fa]]$3],style=point, symbol=[circle,diamond,cross],color=red):\np12 := plot([[2,fb]],style= point,symbol=circle,color=red):\np13 := [plottools[arrow]([.7,-.4],[.7 ,-.15],.02,.15,.3,\n color=COLOR(RGB,.6,0,.9))][ 1]:\np14 := [plottools[arrow]([4.7,-.4],[4.7,-.15],.02,.15,.3,\n \+ color=COLOR(RGB,.6,0,.9))][1]:\np15 := plot([[0.7,-.4 ],[4.7,-.4]],color=COLOR(RGB,.6,0,.9)):\nt1 := plots[textplot]([[0,-.0 5,a],[2,-.05,b]],\n color=COLOR(RGB,0,0,.01)):\nt2 := p lots[textplot]([[.7,-.07,`u=t-2p`],[4.7,-.07,t]],\n col or=blue):\nplots[display]([p1,p2,p3,p4,p5,p6,p7,p8,p9,p10,p11,p12,p13, p14,p15,t1,t2],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }