{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Dark Red Emphasis" -1 259 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Red Emphasis" -1 260 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 261 " Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 260 262 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" 260 265 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 260 266 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 285 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 286 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 287 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 288 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1 " -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 } 1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 24 "The Dirac delta function" }} {PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }} {PARA 0 "" 0 "" {TEXT -1 19 "Version: 27.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 24 "The Dirac delta function" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 91 "A concept which is \+ useful in Fourier analysis is that of an impulse \"function\", called \+ the " }{TEXT 261 20 "Dirac delta function" }{TEXT -1 1 " " }{XPPEDIT 18 0 "delta(x)" "6#-%&deltaG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 84 "It is intended to represent an infinitely narrow, but a lso infinitely high pulse at " }{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 40 "It has the value 0 ever ywhere except at " }{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 24 ", where it is infinite, " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "delta(x)=PIECEWISE([0 ,x<>0 ],[infinity , x=0])" "6#/-% &deltaG6#%\"xG-%*PIECEWISEG6$7$\"\"!0F'F,7$%)infinityG/F'F," }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(delta(x),x = -infinity .. infinity) = Int (delta(x),x = -epsilon .. epsilon);" "6#/-%$IntG6$-%&deltaG6#%\"xG/F*; ,$%)infinityG!\"\"F.-F%6$-F(6#F*/F*;,$%(epsilonGF/F7" }{XPPEDIT 18 0 " ``= 1" "6#/%!G\"\"\"" }{TEXT -1 39 ", for any (small) positive real n umber " }{XPPEDIT 18 0 "epsilon" "6#%(epsilonG" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "It is use d to model a very narrow pulse." }}{PARA 0 "" 0 "" {TEXT -1 80 "The Di rac delta function can be represented graphically by an arrow of heigh t 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 182 "p1 := [plottools[arrow]([0,0],[0,1],.07,.2,.2,color= red)][1]:\np2 := plot(0,x=-2..2,0..2,color=red,thickness=2):\nplots[di splay]([p1,p2],ytickmarks=3,title=`The Dirac delta function`);" }} {PARA 13 "" 1 "" {GLPLOT2D 444 236 236 {PLOTDATA 2 "6(-%)POLYGONSG6&7& 7$$\"+++++N!#6$\"\"!F,7$$!+++++NF*F+7$F.$\"+++++!)!#57$F(F17%7$$!+++++ 5F3F17$F+$\"\"\"F,7$$\"+++++5F3F1-%&STYLEG6#%,PATCHNOGRIDG-%'COLOURG6& %$RGBG$\"*++++\"!\")F+F+-%'CURVESG6%7S7$$!\"#F,F+7$$!3MLLL$Q6G\">!#'***!#=F+7$$!3E++++0\"*H\"*F`pF+7$$!35++++83&H)F`pF+7$$!3\\LLL3k(p`(F `pF+7$$!3Anmmmj^NmF`pF+7$$!3)zmmmYh=(eF`pF+7$$!3+,++v#\\N)\\F`pF+7$$!3 commmCC(>%F`pF+7$$!39*****\\FRXL$F`pF+7$$!3t*****\\#=/8DF`pF+7$$!3=mmm ;a*el\"F`pF+7$$!3komm;Wn(o)!#>F+7$$!3IqLLL$eV(>!#?F+7$$\"3)Qjmm\"f`@') FbrF+7$$\"3%z****\\nZ)H;F`pF+7$$\"3ckmm;$y*eCF`pF+7$$\"3f)******R^bJ$F `pF+7$$\"3'e*****\\5a`TF`pF+7$$\"3'o****\\7RV'\\F`pF+7$$\"3Y'*****\\@f keF`pF+7$$\"3_ILLL&4Nn'F`pF+7$$\"3A*******\\,s`(F`pF+7$$\"3%[mm;zM)>$) F`pF+7$$\"3M*******pfa<*F`pF+7$$\"39HLLeg`!)**F`pF+7$$\"3w****\\#G2A3 \"FTF+7$$\"3;LLL$)G[k6FTF+7$$\"3#)****\\7yh]7FTF+7$$\"3xmmm')fdL8FTF+7 $$\"3bmmm,FT=9FTF+7$$\"3FLL$e#pa-:FTF+7$$\"3!*******Rv&)z:FTF+7$$\"3IL LLGUYo;FTF+7$$\"3_mmm1^rZFTF+7 $$\"\"#F,F+FC-%*THICKNESSG6#F^w-%*AXESTICKSG6$%(DEFAULTG\"\"$-%+AXESLA BELSG6$Q\"x6\"Q!F[x-%&TITLEG6#%9The~Dirac~delta~functionG-%%VIEWG6$;FO F]w;F+F]w" 1 2 0 1 10 0 2 9 1 4 2 1.000000 46.000000 45.000000 0 0 "Cu rve 1" "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "The delta function is " }{TEXT 261 22 "not a genuine function" }{TEXT -1 62 ", \+ but rather, it should be thought of as a limit of functions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "1st des cription of " }{XPPEDIT 18 0 "delta(x)" "6#-%&deltaG6#%\"xG" }{TEXT -1 24 " as a limit of functions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "delta(x)" "6#-%&deltaG6#%\"xG" }{TEXT -1 55 " can be realised as the limit of th e sequence of pulses" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d[n](x) = PIECEWISE([n/2, abs(x) < 1/n],[0, 1/n <= abs(x)]);" "6 #/-&%\"dG6#%\"nG6#%\"xG-%*PIECEWISEG6$7$*&F(\"\"\"\"\"#!\"\"2-%$absG6# F**&F0F0F(F27$\"\"!1*&F0F0F(F2-F56#F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "alias(H =Heaviside):\nd := (n,x) -> n/2*(H(x+1/n)-H(x-1/n)):\nplot([d(1,x),d(2 ,x),d(4,x),d(8,x)],x=-2..2,thickness=2,\n color=[red, green,blue,magenta]);" }}{PARA 13 "" 1 "" {GLPLOT2D 309 279 279 {PLOTDATA 2 "6)-%'CURVESG6$7co7$$!\"#\"\"!F*7$$!1LLL$Q6G\">!#:F*7$$!1n m;M!\\p$=F.F*7$$!1LLL))Qj^.\"F.F*7$$!1LL37&)=@5F.F*7$$!1+Dc^j z:5F.F*7$$!1n;/\">//,\"F.F*7$$!1]7y5\"3x+\"F.F*7$$!1L3_I?,05F.F*7$$!1< /E]fJ-5F.F*7$$!1,+++()>'***!#;$\"1+++++++]Fho7$$!1,+++Y0j&*FhoFio7$$!1 ++++0\"*H\"*FhoFio7$$!1++++83&H)FhoFio7$$!1LLL3k(p`(FhoFio7$$!1nmmmj^N mFhoFio7$$!1ommm9'=(eFhoFio7$$!1,++v#\\N)\\FhoFio7$$!1pmmmCC(>%FhoFio7 $$!1*****\\FRXL$FhoFio7$$!1+++D=/8DFhoFio7$$!1mmm;a*el\"FhoFio7$$!1pmm ;Wn(o)!#!#=Fio7$$\"1Mmm;f`@')F_rFio7$$\"1)****\\nZ)H ;FhoFio7$$\"1lmm;$y*eCFhoFio7$$\"1*******R^bJ$FhoFio7$$\"1'*****\\5a`T FhoFio7$$\"1(****\\7RV'\\FhoFio7$$\"1'*****\\@fkeFhoFio7$$\"1JLLL&4Nn' FhoFio7$$\"1*******\\,s`(FhoFio7$$\"1lmm\"zM)>$)FhoFio7$$\"1*******pfa <*FhoFio7$$\"1km;zy*zd*FhoFio7$$\"1HLLeg`!)**FhoFio7$$\"1m\"H#3Mo+5F.F *7$$\"1+]i5KJ.5F.F*7$$\"1L3-8I%f+\"F.F*7$$\"1mmT:Gd35F.F*7$$\"1L$3-UKQ ,\"F.F*7$$\"1+++D?4>5F.F*7$$\"1LLeM7hH5F.F*7$$\"1mm;W/8S5F.F*7$$\"1LLL j)o61\"F.F*7$$\"1++]#G2A3\"F.F*7$$\"1mm\"H3XL7\"F.F*7$$\"1LLL$)G[k6F.F *7$$\"1++]7yh]7F.F*7$$\"1nmm')fdL8F.F*7$$\"1nmm,FT=9F.F*7$$\"1LL$e#pa- :F.F*7$$\"1+++Sv&)z:F.F*7$$\"1LLLGUYo;F.F*7$$\"1nmm1^rZF.F*7$$\"\"#F*F*-%'COLOURG6&%$RGBG$\"*++++\"!\" )F*F*-F$6$7coF'F+F/F2F5F8F;F>FAFDFGFM7$FfoF*7$F_pF*7$FbpF*7$FepF*7$Fhp F*7$$!1omm;*)o`iFhoF*7$F[qF*7$$!1,+v=My\\cFhoF*7$$!1ML$3P0xU&FhoF*7$$! 1,](oMmmJ&FhoF*7$$!1om\"HKFc?&FhoF*7$$!1,v$4\"y5]^FhoF*7$$!1N$e*)H)e%4 &FhoF*7$$!1^(oHaGo1&FhoF*7$$!1o\"zpyo!R]FhoF*7$$!1&e*)4.48,&FhoF*7$F^q $\"\"\"F*7$$!1NL$3(eR!f%FhoF\\\\l7$FaqF\\\\l7$FdqF\\\\l7$FgqF\\\\l7$Fj qF\\\\l7$F]rF\\\\l7$FarF\\\\l7$FerF\\\\l7$FhrF\\\\l7$F[sF\\\\l7$F^sF\\ \\l7$FasF\\\\l7$$\"1'***\\(3S*eXFhoF\\\\l7$FdsF\\\\l7$$\"1Z7.K?Z#*\\Fh oF\\\\l7$$\"1'\\i!R\\g?]FhoF*7$$\"1ZP4Yyt[]FhoF*7$$\"1(*\\7`2(o2&FhoF* 7$$\"1(\\(=nl8L^FhoF*7$$\"1(**\\7Q-%*=&FhoF*7$$\"1(*\\P4S$>I&FhoF*7$$ \"1(***\\PcY9aFhoF*7$$\"1(**\\P*)G&RcFhoF*7$FgsF*7$$\"1jmmT30piFhoF*7$ FjsF*7$F]tF*7$F`tF*7$FctF*7$FitF*FfvF\\wF_wFbwFewFhwF[xF^xFaxFdxFgxFjx -F^y6&F`yF*FayF*-F$6$7coF'F+F/F2F5F8F;F>FAFDFGFMFgyFhyFiyFjyF[zF_z7$F^ qF*7$FaqF*7$FdqF*7$$!1******\\0zBHFhoF*7$FgqF*7$$!1$eR(**fD'[#FhoF[y7$ $!1m\"zWh\">#FhoF[y7$$!1LL$3ioW3#Fh oF[y7$$!1++v=?=q=FhoF[y7$FjqF[y7$$!1nm;H9Li7FhoF[y7$F]rF[y7$FarF[y7$Fe rF[y7$FhrF[y7$$\"1JL$e*HTW?FhoF[y7$F[sF[y7$$\"1J3_Diu&[#FhoF[y7$$\"1)* \\PMT^7DFhoF*7$$\"1l\"HK/#GRDFhoF*7$$\"1JL3_*\\gc#FhoF*7$$\"1l;zpde>EF hoF*7$$\"1)***\\(e@Jn#FhoF*7$$\"1lm\"HA$>!y#FhoF*7$$\"1KLLe[E()GFhoF*7 $$\"1lm;H\"395$FhoF*7$F^sF*7$$\"1(****\\AYXt$FhoF*7$FasF*7$FdsF*F[_lF_ _lF`_lFa_lFb_lFc_lFfvF\\wF_wFbwFewFhwF[xF^xFaxFdxFgxFjx-F^y6&F`yF*F*Fa y-F$6$7coF'F+F/F2F5F8F;F>FAFDFGFMFgyFhyFiyFjyF[zF_zFi_lFj_lF[`lF_`l7$F falF*7$FjqF*7$$!1mm\"HU8\"f9FhoF*7$F]blF*7$$!1(zJ\"FhoF*7$$\"1(**\\PRi>M\"FhoF*7$$\"1J$eRFV**Q\" FhoF*7$$\"1km;aT#zV\"FhoF*7$$\"1JLe9f)Q`\"FhoF*7$FhrF*7$FdblF*7$F[sF*F bdlFfdlFgdlF[_lF__lF`_lFa_lFb_lFc_lFfvF\\wF_wFbwFewFhwF[xF^xFaxFdxFgxF jx-F^y6&F`yFayF*Fay-%+AXESLABELSG6$Q\"x6\"%!G-%*THICKNESSG6#F\\y-%%VIE WG6$;F(F[y%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "2nd description of " }{XPPEDIT 18 0 "delta(x)" "6#-%&deltaG6#%\"xG" }{TEXT -1 24 " as a limit of func tions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "delta(x)" "6#-%&deltaG6#%\"xG" } {TEXT -1 58 " can be realised as the limit of the sequence of \"wedges \"." }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "t[n](x) = PIE CEWISE([n-n^2*abs(x), abs(x) < 1/n],[0, 1/n <= abs(x)]);" "6#/-&%\"tG6 #%\"nG6#%\"xG-%*PIECEWISEG6$7$,&F(\"\"\"*&F(\"\"#-%$absG6#F*F0!\"\"2-F 46#F**&F0F0F(F67$\"\"!1*&F0F0F(F6-F46#F*" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 191 "ali as(H=Heaviside):\nt := (n,x)->piecewise(x<-1/n,0,x<0,n*(1+n*x),x<1/n,n *(1-n*x),0):\nplot([t(1,x),t(2,x),t(3,x),t(4,x)],x=-2..2,thickness=2, \n color=[red,green,blue,magenta]);" }}{PARA 13 "" 1 "" {GLPLOT2D 401 280 280 {PLOTDATA 2 "6)-%'CURVESG6$7Y7$$!\"#\"\"!F*7$ $!1LLL$Q6G\">!#:F*7$$!1nm;M!\\p$=F.F*7$$!1LLL))Qj^'***!#;$\"1I#*******H,Q!#>7$$!1,+++Y0j&*FS$\"1&** *****RXpV!#<7$$!1++++0\"*H\"*FS$\"1(*******\\*3q)Ffn7$$!1++++83&H)FS$ \"1++++(=\\q\"FS7$$!1LLL3k(p`(FS$\"1nmm\"fBIY#FS7$$!1nmmmj^NmFS$\"1LLL LO[kLFS7$$!1ommm9'=(eFS$\"1KLLL&Q\"GTFS7$$!1,++v#\\N)\\FS$\"1*****\\s] k,&FS7$$!1pmmmCC(>%FS$\"1JLLLvv-eFS7$$!1*****\\FRXL$FS$\"1,++D2YlmFS7$ $!1+++D=/8DFS$\"1+++v\"ep[(FS7$$!1mmm;a*el\"FS$\"1MLL$e/TM)FS7$$!1pmm; Wn(o)Ffn$\"1LLLeDBJ\"*FS7$$!1.++D^bUWFfn$\"1++]([Wdb*FS7$$!1qLLL$eV(>! #=$\"1mmm;kD!)**FS7$$\"1[mmT+07UFfn$\"1NL$e*\\zy&*FS7$$\"1Mmm;f`@')Ffn $\"1PLL3k%y8*FS7$$\"1)****\\nZ)H;FS$\"1-++DB:q$)FS7$$\"1lmm;$y*eCFS$\" 1NLL$o@5a(FS7$$\"1*******R^bJ$FS$\"1,+++'[Wo'FS7$$\"1'*****\\5a`TFS$\" 1/++]*ek%eFS7$$\"1(****\\7RV'\\FS$\"1.++v3mN]FS7$$\"1'*****\\@fkeFS$\" 1/++]ySNTFS7$$\"1JLLL&4Nn'FS$\"1pmmm/\\ELFS7$$\"1*******\\,s`(FS$\"1,+ ++&)ziCFS7$$\"1lmm\"zM)>$)FS$\"1NLL3_;!o\"FS7$$\"1*******pfa<*FS$\"12+ ++ISX#)Ffn7$$\"1km;zy*zd*FS$\"1eLL37-?UFfn7$$\"1HLLeg`!)**FS$\"1(3nm;% RY>Ffr7$$\"1mm;W/8S5F.F*7$$\"1++]#G2A3\"F.F*7$$\"1LLL$)G[k6F.F*7$$\"1+ +]7yh]7F.F*7$$\"1nmm')fdL8F.F*7$$\"1nmm,FT=9F.F*7$$\"1LL$e#pa-:F.F*7$$ \"1+++Sv&)z:F.F*7$$\"1LLLGUYo;F.F*7$$\"1nmm1^rZF.F*7$$\"\"#F*F*-%'COLOURG6&%$RGBG$\"*++++\"!\")F*F*-F$ 6$7coF'F+F/F2F5F8F;F>FAFDFGFJ7$FQF*7$FhnF*7$F]oF*7$FboF*7$FgoF*7$F\\pF *7$$!1ML$3P0xU&FSF*7$Fap$\"1+'*******G!e'Ffr7$$!1NL$3(eR!f%FS$\"1hmm;l TQ;FS7$Ffp$\"1ELLL,.6KFS7$$!1ML$3(3*ew$FS$\"1lmm;lVO\\FS7$F[q$\"1.+++H %=m'FS7$$!1******\\0zBHFS$\"1-+++y$[I)FS7$F`q$\"1,+++F$y%**FS7$$!1LL$3 ioW3#FS$\"1nmm^D@m6F.7$Feq$\"1MLLL=kP8F.7$$!1nm;H9Li7FS$\"1LLLGu1&\\\" F.7$Fjq$\"1LLLBI\\_;F.7$F_r$\"1+++&z(HA=F.7$Fdr$\"1nmmmD5#*>F.7$Fjr$\" 1MLL)*z^J=F.7$F_s$\"1NLLj&Q^l\"F.7$$\"1JLLL1+Y7FS$\"1ommY(*f,:F.7$Fds$ \"1,++I41[8F.7$$\"1JL$e*HTW?FS$\"1omm,[B#=\"F.7$Fis$\"1MLLt'3k,\"F.7$$ \"1KLLe[E()GFS$\"1ummm0%4X)FS7$F^t$\"11+++WzPnFS7$$\"1(****\\AYXt$FS$ \"16+++^\"=1&FS7$Fct$\"1<+++e$eQ$FS7$$\"1'***\\(3S*eXFS$\"1:++]'RUw\"F S7$Fht$\"1E,++]VE9Ffn7$$\"1Z7.K?Z#*\\FS$\"1u8](=(=6IFfr7$$\"1'\\i!R\\g ?]FSF*7$$\"1ZP4Yyt[]FSF*7$$\"1(*\\7`2(o2&FSF*7$$\"1(\\(=nl8L^FSF*7$$\" 1(**\\7Q-%*=&FSF*7$$\"1(*\\P4S$>I&FSF*7$$\"1(***\\PcY9aFSF*7$$\"1(**\\ P*)G&RcFSF*7$F]uF*7$FbuF*7$FguF*7$F\\vF*7$FavF*7$F[wF*FbwFewFhwF[xF^xF axFdxFgxFjxF]yF`yFcy-Fgy6&FiyF*FjyF*-F$6$7_pF'F+F/F2F5F8F;F>FAFDFGFJF` zFazFbzFczFdzFez7$FapF*7$FfpF*7$Fe[lF*7$F[qF*7$$!1****\\7\\;HJFS$\"13+ ](y:v$=FS7$F]\\l$\"1/++]])eo$FS7$$!1++](=;%=FFS$\"1/+]7VDMbFS7$F`q$\"1 +++vNi#Q(FS7$$!1mm\"HAb()H#FS$\"1.+v$*H?6$*FS7$Fe\\l$\"1++DT#yR7\"F.7$ $!1++v=?=q=FS$\"1+]7$=OoJ\"F.7$Feq$\"1+++DTp4:F.7$$!1mm\"HU8\"f9FS$\"1 +]P>zz'o\"F.7$F]]l$\"1++v8BFfn$\"1+](oS+7z#F.7$Fdr$\"1+++v2B#)HF.7$$\"1cm;/rI2?Ffn$\" 1,]igBM>GF.7$Fjr$\"1-+D'\\:4i#F.7$$\"1Tm;zHz;kFfn$\"1.](=j)[ACF.7$F_s$ \"1.+]n<1CAF.7$$\"1(***\\7r2a5FS$\"1.+v)fI80#F.7$Fa^l$\"1-++I%*fy=F.7$ $\"1km;aT#zV\"FS$\"1-+Dh#oeq\"F.7$Fds$\"1-+]#4PJ`\"F.7$$\"1lmTN.8P=FS$ \"1-]7)p#eY8F.7$Fi^l$\"1-+v.$G+;\"F.7$$\"1)**\\il&p^AFS$\"1=+v$4RZt*FS 7$Fis$\"1>++]^>pyFS7$$\"1)***\\(e@Jn#FS$\"1<+]7d!>%fFS7$Fa_l$\"1;++vih 9SFS7$$\"1lm;H\"395$FS$\"19+]PoK(3#FS7$F^t$\"1F,++SP+;Ffn7$$\"1*\\ildQ FAFD FGFJF`zFazFbzFczFdzFezF`clFaclFccl7$F]\\lF*7$F`qF*7$Fedl$\"1RLLLk\"*>K FS7$Fe\\l$\"1tmmm?][mFS7$F]el$\"1,++q(3x+\"F.7$Feq$\"1MLLLtc]8F.7$Feel $\"1MLLB&=am\"F.7$F]]l$\"1MLL8(p-)>F.7$F]fl$\"1LLL.47&H#F.7$Fjq$\"1LLL $4s*4EF.7$Fefl$\"1mmmO;e\\HF.7$F_r$\"1+++!=\">*G$F.7$F]gl$\"1LLLB2!)GO F.7$Fdr$\"1mmmm-ToRF.7$Fegl$\"1NLLj3$)yOF.7$Fjr$\"1OLL$*>2ELF.7$F]hl$ \"1PLLBJJtHF.7$F_s$\"1RLL`Ub?EF.7$Fehl$\"10++?mZ8BF.7$Fa^l$\"1rmm')*)R 1?F.7$F]il$\"1PLL`8K*p\"F.7$Fds$\"1.++?PC#R\"F.7$Feil$\"1PLLj9fg5F.7$F i^l$\"1+nmm?R*G(FS7$F]jl$\"1M+++&pG(RFS7$Fis$\"1rOLL$pMc'Ffn7$$\"1J3_D iu&[#FS$\"1,qm;Rg!G#Ffn7$$\"1)*\\PMT^7DFSF*7$$\"1l\"HK/#GRDFSF*7$$\"1J L3_*\\gc#FSF*7$$\"1l;zpde>EFSF*7$FejlF*7$$\"1lm\"HA$>!y#FSF*7$Fa_lF*7$ F][mF*7$F^tF*Fi\\mF]]mF^]mFeblFfblFgblFhblFiblFjblFbwFewFhwF[xF^xFaxFd xFgxFjxF]yF`yFcy-Fgy6&FiyFjyF*Fjy-%*THICKNESSG6#Fey-%+AXESLABELSG6$Q\" x6\"%!G-%%VIEWG6$;F(Fdy%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "3rd description of " } {XPPEDIT 18 0 "delta(x)" "6#-%&deltaG6#%\"xG" }{TEXT -1 24 " as a limi t of functions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "delta(x)" "6#-%&deltaG6 #%\"xG" }{TEXT -1 59 " can be realised as the limit of the sequence of functions " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "h[n](x ) = n/(Pi*(1+n^2*x^2))" "6#/-&%\"hG6#%\"nG6#%\"xG*&F(\"\"\"*&%#PiGF,,& F,F,*&F(\"\"#F*F1F,F,!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "Int(n/(Pi*(1+n^2*x^2)),x) = arctan* n*x/Pi;" "6#/-%$IntG6$*&%\"nG\"\"\"*&%#PiGF),&F)F)*&F(\"\"#%\"xGF.F)F) !\"\"F/**%'arctanGF)F(F)F/F)F+F0" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "Int(n/(Pi*(1+n^2*x^2)),x=-infinity..infinity)=1" "6#/-%$IntG6$*& %\"nG\"\"\"*&%#PiGF),&F)F)*&F(\"\"#%\"xGF.F)F)!\"\"/F/;,$%)infinityGF0 F4F)" }{TEXT -1 27 ", for any positive integer " }{TEXT 267 1 "n" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Int(n/(Pi*(1+n^2*x^2)),x);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&%\"nG\"\"\"*&%#PiGF(,&F(F(*&)F'\" \"#F()%\"xGF.F(F(F(!\"\"F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%'arc tanG6#*&%\"nG\"\"\"%\"xGF)F)%#PiG!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "h := (n,x) -> n/(Pi*(1 +n^2*x^2));\nplot([h(1,x),h(2,x),h(3,x),h(4,x)],x=-4..4,\n c olor=[red,green,blue,magenta]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"hGf*6$%\"nG%\"xG6\"6$%)operatorG%&arrowGF)*&9$\"\"\"*&%#PiGF/,&F/F/* &)F.\"\"#F/)9%F5F/F/F/!\"\"F)F)F)" }}{PARA 13 "" 1 "" {GLPLOT2D 360 270 270 {PLOTDATA 2 "6(-%'CURVESG6$7ao7$$!\"%\"\"!$\"1p()>&46C(=!#<7$$ !1nmmmFiDQ!#:$\"1lxf%eHe.#F-7$$!1LLLo!)*Qn$F1$\"1]c!y0:c>#F-7$$!1nmmwx E.NF1$\"1_c(*\\m>)R#F-7$$!1mmmOk]JLF1$\"1jpW[<*3j#F-7$$!1MLL[9cgJF1$\" 18wStOe'*GF-7$$!1nmmhN2-IF1$\"1h\"ok?U\"zJF-7$$!1+++N&oz$GF1$\"1?#3<-d c^$F-7$$!1nmm\")3DoEF1$\"1z%pHe$G?RF-7$$!1+++:v2*\\#F1$\"1e3!*fcF$R%F- 7$$!1LLL8>1DBF1$\"1b)*Qd6+p\\F-7$$!1nmmw))yr@F1$\"1,k(Qr-\"obF-7$$!1++ +S(R#**>F1$\"1Rjj:82qjF-7$$!1++++@)f#=F1$\"1u3,=f7WtF-7$$!1+++gi,f;F1$ \"1mjhs;)H[)F-7$$!1nmm\"G&R2:F1$\"1'>7$\\@eF(*F-7$$!1LLLtK5F8F1$\"1-E? !=%z_6!#;7$$!1MLL$HsV<\"F1$\"17LxjT\"zL\"Fip7$$!1-++]&)4n**Fip$\"1G*yN Z%z'f\"Fip7$$!1qmmT.AFip7$$!1*******4\"eZeFip$\"1Im3&=4?P#Fip7$$!1******\\O3E]Fip$\"1N> bME;TDFip7$$!1mmmTs$*oTFip$\"1z8&Q#*)y6FFip7$$!1KLLL3z6LFip$\"1u^laY[o GFip7$$!1LLLeGmCDFip$\"1O7akwO#*HFip7$$!1MLL$)[`PL$3Fr)4= F-$\"1*Qul`c?=$Fip7$$\"16LL3Uh9SF-$\"16'pg(o(z<$Fip7$$\"1/L$e9d$>iF-$ \"16SC[R$3<$Fip7$$\"1(HLL3+TU)F-$\"1A-*z-p1;$Fip7$$\"1GL$efeLG\"Fip$\" 1!))yrRA:8$Fip7$$\"1FLL$=2Vs\"Fip$\"1B0)f\\!>\"4$Fip7$$\"1hmmm7+#\\#Fi p$\"1@-NXS)p*HFip7$$\"1'*****\\`pfKFip$\"1oT\"z`ht(GFip7$$\"1imm\"*f#) )3%Fip$\"1oGEi)ers#Fip7$$\"1HLLLm&z\"\\Fip$\"14jC;V;jDFip7$$\"1jmm;(HX x&Fip$\"1?!=2\"=6(Q#Fip7$$\"1(******z-6j'Fip$\"1NfODD#4@#Fip7$$\"1%*** **\\C4puFip$\"1@o$zILK/#Fip7$$\"1#******4#32$)Fip$\"1BT?3cS$)=Fip7$$\" 1$****\\qM8F1$\"1xAJc7TW 6Fip7$$\"1++++.W2:F1$\"1-#oYpyrs*F-7$$\"1LLLep'Rm\"F1$\"1#*oP+F&fW)F-7 $$\"1+++S>4N=F1$\"1xVBOV/)G(F-7$$\"1mmm6s5'*>F1$\"1\"zy<0mgQ'F-7$$\"1+ ++lXTk@F1$\"1Q/,@MU*f&F-7$$\"1mmmmd'*GBF1$\"1Qf]'yZ\\&\\F-7$$\"1+++DcB ,DF1$\"1x%*)*z?u'Q%F-7$$\"1MLLt>:nEF1$\"1z<&oX;J#RF-7$$\"1LLL.a#o$GF1$ \"1]y&yrx\"=NF-7$$\"1nmm^Q40IF1$\"1\"G0cv\"RtJF-7$$\"1+++!3:(fJF1$\"1* e7\\N%*z*GF-7$$\"1nmmc%GpL$F1$\"13)H?.`Ii#F-7$$\"1LLL8-V&\\$F1$\"1:!ee y[\"3CF-7$$\"1+++XhUkOF1$\"1dE5&f#>1AF-7$$\"1+++:o6()pd6F-7$F:$\"1lU)fu74F\"F-7$ F?$\"10vRmxP-9F-7$FD$\"1R&G'=nPa:F-7$FI$\"1\"R#**o?G=F-7$FS$\"1$*[X9`if@F-7$FX$\"1ll]!*fF]CF-7$Fgn$\"12yKan&R\"GF-7$F\\o$ \"1WJ*[xhW?$F-7$Fao$\"1bpukJ]ZPF-7$Ffo$\"1;XM\"zF-7$F[q$\"1c-$yg*>p(*F-7$F` q$\"1$>#y+l'*z7Fip7$Feq$\"1dJ>ZMIc9Fip7$Fjq$\"1E1)4R8rm\"Fip7$F_r$\"1z $>i!)zt%>Fip7$Fdr$\"1([8^3%Fip7$Fhs$\"1Qi*=**3\\U%Fip7$$!1LL$e%oA=HFip$ \"1'[1L.>'[ZFip7$F]t$\"1'eFerTG2&Fip7$$!1LL$3())4J@Fip$\"1`+7f()[(Q&Fi p7$Fbt$\"1IS^&3X-o&Fip7$F\\u$\"1oJvY$>8<'Fip7$F`v$\"1;r;#p+eO'Fip7$Few $\"1;%R,zt/>'Fip7$F_x$\"1-IA^Oa*o&Fip7$$\"1%****\\Aa\"3@Fip$\"1#3e77'G 0aFip7$Fdx$\"1X?zJSZ*4&Fip7$$\"1HLL3$[e(GFip$\"1?s1+]m$y%Fip7$Fix$\"1N o>1-VnWFip7$$\"1HL$3ngUn$Fip$\"1]6@mS(Q8%Fip7$F^y$\"1bh#=*H(\\\"QFip7$ $\"1'***\\78R.XFip$\"13t\\)ek[^$Fip7$Fcy$\"1+Xj[xvNKFip7$Fhy$\"1M3oVe \"ys#Fip7$F]z$\"1KJq!oXvI#Fip7$Fbz$\"15Pl=![+(>Fip7$Fgz$\"1r,9x0+$p\"F ip7$F\\[l$\"11uC_q!=Z\"Fip7$Fa[l$\"1$yJ`z$)yG\"Fip7$Ff[l$\"1=gQfkq*y*F -7$F[\\l$\"1l$32@YCF-7$Fc^l$\"1IB[\\[Mh@F-7$Fh^l$\"1%f A=1*3=>F-7$F]_l$\"182!z&G#\\r\"F-7$Fb_l$\"15ms-#*=b:F-7$Fg_l$\"1\"oqE9 CzR\"F-7$F\\`l$\"1^h#*)3/lF\"F-7$Fa`l$\"1K&zs#=ej6F-7$Ff`l$\"1uo5vz*)o 5F-7$F[alFhal-F^al6&F`alF*FaalF*-F$6$7co7$F($\"16H6$y@de'Fbv7$F/$\"1H \\fb2:&>(Fbv7$F5$\"1,G&Qtgnz(Fbv7$F:$\"1nx%=&4#p;F-7$Fgn$\"1fY'))*z>B>F-7$F\\o$\"1NsaXmw( >#F-7$Fao$\"1*oVg8,Ge#F-7$Ffo$\"1()G0uQjzIF-7$F[p$\"1uUj&oSaq$F-7$F`p$ \"19X28B&=X%F-7$Fep$\"1Thq(*)fpm&F-7$F[q$\"1;o,'\\x(>rF-7$F`q$\"1uyN'y #41'*F-7$Fjq$\"1urid$G1I\"Fip7$F_r$\"1G$*p2#\\Sc\"Fip7$Fdr$\"1bV.JUv3> Fip7$Fir$\"1m\">'y7'>M#Fip7$F^s$\"1$3M)4$>r\"HFip7$Fcfl$\"1=TnMp?!H$Fi p7$Fcs$\"1mb6A#zSs$Fip7$F[gl$\"1QhH*Q\")pA%Fip7$Fhs$\"1n5hg`g0[Fip7$Fc gl$\"1')Qz7E%fS&Fip7$F]t$\"1Ds\"R7N#ogFip7$F[hl$\"1K\"=ai'fynFip7$Fbt$ \"1O_:+g+4vFip7$Fgt$\"1>HrJ_im#)Fip7$F\\u$\"1cC=I:#e\"*)Fip7$Ffu$\"1= \"3'p6yn$*Fip7$F`v$\"1!fC'3n&za*Fip7$F[w$\"1J5Q\"fgFT*Fip7$Few$\"1aA^V '3g(*)Fip7$Fjw$\"1X(GFMIlJ)Fip7$F_x$\"1IzI5*>M`(Fip7$F_il$\"1tKnFK)4#o Fip7$Fdx$\"1qGuw%Qc7'Fip7$Fgil$\"1nur_,VuaFip7$Fix$\"1(3'pGFip7$Fbz$\"1We q3^.'e\"Fip7$Fgz$\"1cfeCeKC8Fip7$Fa[l$\"1;,k[`.t'*F-7$Ff[l$\"104QF*=h8 (F-7$F[\\l$\"1\"=v'*e&R1cF-7$F`\\l$\"1'4Q8\"))f^WF-7$Fe\\l$\"1NK'QxtUo $F-7$Fj\\l$\"1>o'*py5]IF-7$F_]l$\"1?%3V(Qp!f#F-7$Fd]l$\"1%y6z[?C@#F-7$ Fi]l$\"1(*[x>`)o\">F-7$F^^l$\"1m^ze0Qm;F-7$Fc^l$\"1E(Hg>F-7$Ffo$\"1!>3)>/yUBF-7$F[p$\"1j3/gT2FGF- 7$F`p$\"1.(o;]2%3MF-7$Fep$\"1:QAY-^jVF-7$F[q$\"1.rtWt))>bF-7$F`q$\"1y_ _a6COvF-7$Fjq$\"1l=(z\"4GP5Fip7$Fdr$\"1LC$H0_(o:Fip7$Fir$\"1?kC3len>Fi p7$F^s$\"1R6)\\jX`_#Fip7$Fcs$\"1QWGy/knLFip7$Fhs$\"1!zy,fr#*Q&Fip7$F]t$\"1.-DQSq.jFip7$F[hl$\"1T(f#z<.utFip7$Fbt$\"15A &)4(4`e)Fip7$$!1++v=*y__\"Fip$\"1,f\"=uu&y#*Fip7$Fgt$\"1*=-(e5ez**Fip7 $$!1MLe*)pw+6Fip$\"16JLF3[m5F17$F\\u$\"1_&=C$)\\/8\"F17$Ffu$\"1:7./0%3 B\"F17$F`v$\"19P%*y>#HF\"F17$F[w$\"1NaV49BT7F17$Few$\"1^?-;7TV6F17$$\" 1HL3-V)G1\"Fip$\"1_xE!*fKy5F17$Fjw$\"1Dxa\\7p25F17$$\"1GLe*)G$Q]\"Fip$ \"1!)=I/;R\\$*Fip7$F_x$\"1Y&ewpMzi)Fip7$F_il$\"1BK+C45TuFip7$Fdx$\"1Hb H:*)f'Q'Fip7$Fgil$\"1(3C>3_.[&Fip7$Fix$\"1cL,00`:ZFip7$F^y$\"1X)GTTNYY $Fip7$Fcy$\"1qMo:#eXh#Fip7$Fhy$\"1vsX^gx4?Fip7$F]z$\"1_P5&>HXe\"Fip7$F gz$\"1Erj]4Sd5Fip7$Fa[l$\"1\\,F\"z%>\"f(F-7$Ff[l$\"1&G[O*[)H`&F-7$F[\\ l$\"1!*[L\\ok:VF-7$F`\\l$\"1,h-w$4#3MF-7$Fe\\l$\"1)oIbaZ1\"GF-7$Fj\\l$ \"1K!ymn++K#F-7$F_]l$\"16UC#ffj'>F-7$Fd]l$\"10%)z8nIw;F-7$Fi]l$\"1gjU# p,/X\"F-7$F^^l$\"1&)[q;.Sf7F-7$Fc^l$\"1*4UBF3*36F-7$Fh^l$\"1@n#zZq@\") *Fbv7$F]_l$\"1%Qu5JB9v)Fbv7$Fb_l$\"1&)y+sI2@zFbv7$Fg_l$\"15TI@'fm5(Fbv 7$F\\`l$\"1,^-*>p*zkFbv7$Fa`l$\"1;Sc=-x)*eFbv7$Ff`l$\"1:bg<&\\ET&Fbv7$ F[alFb]n-F^al6&F`alFaalF*Faal-%+AXESLABELSG6$Q\"x6\"%!G-%%VIEWG6$;F(F[ al%(DEFAULTG" 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" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Int(h(10^6,x),x=-1/10 ^3..1/10^3);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$ *&\"\"\"F(*&%#PiGF(,&F(F(*$)%\"xG\"\"#F(\".++++++\"F(!\"\"\"(+++\"/F.; #F1\"%+5#F(F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+/Qj$***!#5" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "4th description of " } {XPPEDIT 18 0 "delta(x)" "6#-%&deltaG6#%\"xG" }{TEXT -1 24 " as a limi t of functions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "delta(x)" "6#-%&deltaG6 #%\"xG" }{TEXT -1 59 " can be realised as the limit of the sequence of functions " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "p[n](x ) = n*exp(-n*abs(x))/2;" "6#/-&%\"pG6#%\"nG6#%\"xG*(F(\"\"\"-%$expG6#, $*&F(F,-%$absG6#F*F,!\"\"F,\"\"#F5" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 9 "Note that" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "Int(p[n](x),x = -infinity .. infinity) = Int(n*exp(n*x)/2,x = -i nfinity .. 0)+Int(n*exp(-n*x)/2,x = 0 .. infinity);" "6#/-%$IntG6$-&% \"pG6#%\"nG6#%\"xG/F-;,$%)infinityG!\"\"F1,&-F%6$*(F+\"\"\"-%$expG6#*& F+F7F-F7F7\"\"#F2/F-;,$F1F2\"\"!F7-F%6$*(F+F7-F96#,$*&F+F7F-F7F2F7F " 0 "" {MPLTEXT 1 0 117 "assume(n_,posint):\nInt(n*exp(n*x)/2,x=-infinity..0) +Int(n*exp(-n*x)/2,x=0..infinity);\nsubs(n_=n,value(subs(n=n_,%)));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$IntG6$,$*&%\"nG\"\"\"-%$expG6#*& F)F*%\"xGF*F*#F*\"\"#/F/;,$%)infinityG!\"\"\"\"!F*-F%6$,$*&F)F*-F,6#,$ F.F6F*F0/F/;F7F5F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "p := (n,x) -> piecewise(x<=0,n*exp(n*x)/2,x>0,n*exp(-n*x)/2):\npl ot([p(1,x),p(2,x),p(3,x),p(4,x)],x=-4..4,\n color =[red,green,blue,magenta]);;" }}{PARA 13 "" 1 "" {GLPLOT2D 360 270 270 {PLOTDATA 2 "6(-%'CURVESG6$7fn7$$!\"%\"\"!$\"1*3nVW>y:*!#=7$$!1nmm mFiDQ!#:$\"1([S$)fU-4\"!#<7$$!1LLLo!)*Qn$F1$\"1ayc6x')o7F47$$!1nmmwxE. NF1$\"1Q:c3L%\\]\"F47$$!1mmmOk]JLF1$\"1?uu!Ghpy\"F47$$!1MLL[9cgJF1$\"1 UDA&R'4?@F47$$!1nmmhN2-IF1$\"1%3d&Qp>%[#F47$$!1+++N&oz$GF1$\"1*QL\\\"Q AFHF47$$!1nmm\")3DoEF1$\"1W[=#3t'oMF47$$!1+++:v2*\\#F1$\"1.U/(yP!3TF47 $$!1LLL8>1DBF1$\"1W(*)p]p)))[F47$$!1nmmw))yr@F1$\"1Hqz!\\x')p&F47$$!1+ ++S(R#**>F1$\"1:o-=1\">x'F47$$!1++++@)f#=F1$\"1VUC+&pH0)F47$$!1+++gi,f ;F1$\"1rtusfI;&*F47$$!1nmm\"G&R2:F1$\"1g9M\"oIu5\"!#;7$$!1LLLtK5F8F1$ \"1-ji^CAE8Fep7$$!1MLL$HsV<\"F1$\"1gUOTW1X:Fep7$$!1-++]&)4n**Fep$\"1w= 4c!fa%=Fep7$$!1PLLL\\[%R)Fep$\"1:nj,Muf@Fep7$$!1)*****\\&y!pmFep$\"1Zn s=lYmDFep7$$!1*******4\"eZeFep$\"1XPC#=.iy#Fep7$$!1******\\O3E]Fep$\"1 i'4$RLvCIFep7$$!1mmmTs$*oTFep$\"1&)oJ3[X&H$Fep7$$!1KLLL3z6LFep$\"1yX!* 4GQ!f$Fep7$$!1LLLeGmCDFep$\"1&H.<-7W)QFep7$$!1MLL$)[`P'p(H33UFe p7$$\"1hmmm7+#\\#Fep$\"10xEr)>r*QFep7$$\"1'*****\\`pfKFep$\"1r&3y)e84O Fep7$$\"1imm\"*f#))3%Fep$\"1k[sR6'>K$Fep7$$\"1HLLLm&z\"\\Fep$\"1aRf_lj dIFep7$$\"1jmm;(HXx&Fep$\"1c\\?18j1GFep7$$\"1(******z-6j'Fep$\"1n4^(RJ id#Fep7$$\"1%*****\\C4puFep$\"1^$yF!R9pBFep7$$\"1#******4#32$)Fep$\"1I q\"f!Hqy@Fep7$$\"1%*****\\#y'G**Fep$\"1Z&*>.Ic_=Fep7$$\"1******H%=H<\" F1$\"1\\0h(Q7ta\"Fep7$$\"1mmm1>qM8F1$\"1!)[BkJ=;8Fep7$$\"1++++.W2:F1$ \"1v+yF3Q26Fep7$$\"1LLLep'Rm\"F1$\"1I&e*R+Jp%*F47$$\"1+++S>4N=F1$\"1=u ')y8%*zzF47$$\"1mmm6s5'*>F1$\"1'*p5!3dJz'F47$$\"1+++lXTk@F1$\"1*)=nWm& 3u&F47$$\"1mmmmd'*GBF1$\"1QvR#G@)p[F47$$\"1+++DcB,DF1$\"1BZ?\\<=*4%F47 $$\"1MLLt>:nEF1$\"1Q=4Jp[sMF47$$\"1LLL.a#o$GF1$\"1:bZI>dIHF47$$\"1nmm^ Q40IF1$\"1Qvfi_qwCF47$$\"1+++!3:(fJF1$\"1$fVs`\"*=7#F47$$\"1nmmc%GpL$F 1$\"1(*yMg&)Hx^-ziiaL!#>7$F/$\"1p[%Hp:Xv%Fc ^l7$F6$\"1tx$y2,,W'Fc^l7$F;$\"1#fOzW<%f!*Fc^l7$F@$\"1$=cvC#Hx7F-7$FE$ \"1ZU,!\\Bzz\"F-7$FJ$\"1cX\"=x$\\oCF-7$FO$\"1u_30dXFMF-7$FT$\"1\"[H/=x E\"[F-7$FY$\"1:K\"Ry*Q]nF-7$Fhn$\"1f'\\D-=/c*F-7$F]o$\"1pYw0q**)H\"F47 $Fbo$\"1tG(HlFep7$Fis$\"1!RlE&3ZkqFep7$$!1nm;aH-88F ep$\"1d'y+Df/p(Fep7$F^t$\"16Esyp\">P)Fep7$$!1SLLe4**RYF4$\"1dTG?#fP6*F ep7$Fct$\"1%>?YpO8#**Fep7$$\"16LL3Uh9SF4$\"1`o@NdYG#*Fep7$Fht$\"1UJ54Z Y\\%)Fep7$$\"1GL$efeLG\"Fep$\"1)*)RSP@it(Fep7$F]u$\"1&p)RO\\=$3(Fep7$$ \"1%****\\Aa\"3@Fep$\"1rYy::wflFep7$Fbu$\"1X=T;t,vgFep7$$\"1HLL3$[e(GF ep$\"1wOfuS4EcFep7$Fgu$\"10rmUZM5_Fep7$F\\v$\"1[o`D.<9WFep7$Fav$\"1dM5 :llRPFep7$Ffv$\"1&*ye:<(3:$Fep7$F[w$\"1C&3aG(yaEFep7$F`w$\"1M(GG8P^C#F ep7$Few$\"1C0!4a)p)*=Fep7$Fjw$\"1\"\\HB(ezs8Fep7$F_x$\"1GAT*\\-nd*F47$ Fdx$\"1o8E([_$HpF47$Fix$\"1O&R4>p^!\\F47$F^y$\"1wVJ0Lr'e$F47$Fcy$\"1mK#[F-7$Fa[l$\"1%Rwet+`V$F-7$Ff[l$\" 1O86$eFOX#F-7$F[\\l$\"1\"RI$y%p4!=F-7$F`\\l$\"1$)e/pg^j7F-7$Fe\\l$\"1, e:&zVD?*Fc^l7$Fj\\l$\"1q!fUFkKc'Fc^l7$F_]l$\"1n\"fy9]#\\ZFc^l7$Fd]lFa^ l-Fg]l6&Fi]lF*Fj]lF*-F$6$7ao7$F($\"19B**H&=j@*!#@7$F/$\"1_!yGms]b\"!#? 7$F6$\"1102&*3\\^CFf\\m7$F;$\"1V8K+HILJN9\"Fc^l7$FJ$\"1%oO7Sn'R=Fc^l7$FO$\"1xkO*)*y)4IFc^l7$FT$\"1* H+n8\"33]Fc^l7$FY$\"1*yxG*oD>$)Fc^l7$Fhn$\"1ZbL%3*=-9F-7$F]o$\"1d\"4#* \\p2A#F-7$Fbo$\"19Frv\"=ms$F-7$Fgo$\"1=c(yH;F47$Fgp$\"1tRJ!3x\"*z#F47$F\\q$\"1e'Gk8)4EWF47$Faq$ \"1b')3[#Q@a(F47$Ffq$\"1C2[%e#*)37Fep7$F[r$\"1z\\r@4cG?Fep7$F`r$\"1m=T @9\\&f#Fep7$Fer$\"1e>W&Qk3K$Fep7$Fjr$\"1jM(=cXYH%Fep7$F_s$\"1y'Q')zpRb &Fep7$Fbcl$\"1Ne_sU+]iFep7$Fds$\"1XcbirELqFep7$Fjcl$\"1!\\mi5!p9zFep7$ Fis$\"16?i:Zd1*)Fep7$Fbdl$\"1#Q)>w_i65F17$F^t$\"1/jU_F-\\6F17$$!1tm;/1 binF4$\"1y_#[coXA\"F17$Fjdl$\"15sQa8308F17$$!12+]78VL$3Fr)4=F4$\"1Z4\"fOF2U\"F17$Fbel$\"1 \\m64vzH8F17$$\"1/L$e9d$>iF4$\"1S*3ZP(oW7F17$Fht$\"1u_\"p[C];\"F17$Fje l$\"12$)))=#o1-\"F17$F]u$\"1Y(fg)*))>%*)Fep7$Fbfl$\"1+y.pzNpzFep7$Fbu$ \"1,_i57_-rFep7$Fjfl$\"1g)[1Mr*HjFep7$Fgu$\"1oVefEXTcFep7$F\\v$\"1=!oc o5\"*R%Fep7$Fav$\"1OzTOJNIMFep7$Ffv$\"179kf0+`EFep7$F[w$\"1/Z^2R!=0#Fe p7$Few$\"1>hb3+,T7Fep7$Fjw$\"1*)z1KDdHwF47$F_x$\"1L/%H7XaW%F47$Fdx$\"1 _Z$[7*3OFF47$Fix$\"1$=>d4n&H;F47$F^y$\"1))RP45\"*=5F47$Fcy$\"1_!pjr+z4 'F-7$Fhy$\"1?&z3Y/=w$F-7$F]z$\"1bRa**)3YT;Fb\\m7$F@$\"1_UL r4&HE$Fb\\m7$FE$\"1?+*=vd]Y'Fb\\m7$FJ$\"1^uH+Bp=7Ff\\m7$FO$\"1=V1A0\\ \\BFf\\m7$FT$\"1J`gGBPKYFf\\m7$FY$\"1!)>&pW_N6*Ff\\m7$Fhn$\"1ZyJb=.G=F c^l7$F]o$\"1]6(>W'yuLFc^l7$Fbo$\"1R3D=noHnFc^l7$Fgo$\"1ed$))Q!yX8F-7$F \\p$\"1&H$)>&4OCEF-7$Fap$\"1ie\"Rd.I\"[F-7$Fgp$\"1NP(*R1b**)*F-7$F\\q$ \"15#*)=\\GO#=F47$Faq$\"1M_f)Gb;r$F47$$!1qmmTFep7$Fer$\"1e$p/<7'yEFep7$Fjr$\"18Z]kF3uPFep7$F_s$\"1M(pWQn vJ&Fep7$Fbcl$\"1z&oA$*yTA'Fep7$Fds$\"16>#[+i`G(Fep7$Fjcl$\"11+wn+ZF&)F ep7$Fis$\"1rF17$F_cm$\"1h/)[4F.'=F17$Fbel$\"1J9_'f\"H.^f:F17$Fht$\"1*e2u2pyU\"F17$$\"1HL3-V)G1\"Fep$\"1\">N'*eQtI\"F17$Fj el$\"1NI'HU#)p>\"F17$$\"1GLe*)G$Q]\"Fep$\"1+\\-M<%f4\"F17$F]u$\"13X'o< IM+\"F17$Fbfl$\"1@RzGU41')Fep7$Fbu$\"1b2%*yq;\"Q(Fep7$Fjfl$\"1;cB2peIj Fep7$Fgu$\"1Zx[n%Q&HaFep7$F\\v$\"1;+dK*zp*QFep7$Fav$\"1Y6>qh+(z#Fep7$F fv$\"1h$=O^)f&)>Fep7$F[w$\"1_L:1\"z&49Fep7$F`w$\"1'GT!\\\"G\"35Fep7$Fe w$\"1gbV)H7,@(F47$$\"1$****\\'4a9oFc^l7$F]z$\"1Q&)y sg\"eZ$Fc^l7$Fbz$\"1uuoNzFl%Ff\\m7$Fa[l$\"1zD=H#e-O#Ff\\m7$Ff[l$\"1c4Kjw0/7Ff\\m7$F[\\l$ \"1!GbP7#)p['Fb\\m7$F`\\l$\"1>VZ8d%H>$Fb\\m7$Fe\\l$\"15_.Yit$p\"Fb\\m7 $Fj\\l$\"1(3`ne(G:')F^jm7$F_]l$\"1j!)Q$Rv5^%F^jm7$Fd]lF\\jm-Fg]l6&Fi]l Fj]lF*Fj]l-%+AXESLABELSG6$Q\"x6\"%!G-%%VIEWG6$;F(Fd]l%(DEFAULTG" 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" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "Int('p(10^4,x)',x=-1/10^3..1/10^3); \nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%\"pG6$\"&++ \"%\"xG/F*;#!\"\"\"%+5#\"\"\"F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" +,ga****!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "5th des cription of " }{XPPEDIT 18 0 "delta(x)" "6#-%&deltaG6#%\"xG" }{TEXT -1 24 " as a limit of functions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {XPPEDIT 18 0 "delta(x)" "6#-%&deltaG6#% \"xG" }{TEXT -1 59 " can be realised as the limit of the sequence of f unctions " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[n](x) \+ = n/sqrt(2*Pi);" "6#/-&%\"bG6#%\"nG6#%\"xG*&F(\"\"\"-%%sqrtG6#*&\"\"#F ,%#PiGF,!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-n^2*x^2/2);" "6#-% $expG6#,$*(%\"nG\"\"#%\"xGF)F)!\"\"F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "Int(exp(-n^2*x^2/2),x = -i nfinity .. infinity) = sqrt(2*Pi)/n;" "6#/-%$IntG6$-%$expG6#,$*(%\"nG \"\"#%\"xGF-F-!\"\"F//F.;,$%)infinityGF/F3*&-%%sqrtG6#*&F-\"\"\"%#PiGF 9F9F,F/" }{TEXT -1 9 ", so that" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "n/sqrt(2*Pi);" "6#*&%\"nG\"\"\"-%%sqrtG6#*&\"\"#F%%#PiG F%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(exp(-n^2*x^2/2),x = -infi nity .. infinity) = 1;" "6#/-%$IntG6$-%$expG6#,$*(%\"nG\"\"#%\"xGF-F-! \"\"F//F.;,$%)infinityGF/F3\"\"\"" }{TEXT -1 27 ", for any positive in teger " }{TEXT 269 1 "n" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "Int(exp(-n^2*x^2/2),x=- infinity..infinity);\nassume(n_>0):\nsubs(n_=n,value(subs(n=n_,%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%$expG6#,$*&)%\"nG\"\"#\" \"\")%\"xGF-F.#!\"\"F-/F0;,$%)infinityGF2F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&*&-%%sqrtG6#\"\"#\"\"\"-F&6#%#PiGF)F)%\"nG!\"\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "b := (n,x) -> n*exp(-n^2*x^2/2)/sqrt(2*Pi):\nplot([b(1,x),b(2,x), b(3,x),b(4,x)],x=-4..4,\n color=[red,green,blue,magenta ]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 270 270 {PLOTDATA 2 "6(-%'CURVES G6$7]o7$$!\"%\"\"!$\"1a)[wD-$Q8!#>7$$!1nmmmFiDQ!#:$\"1wC4K;uZEF-7$$!1L LLo!)*Qn$F1$\"1s6w]K\"pn%F-7$$!1nmmwxE.NF1$\"1P)p#p)Qvi)F-7$$!1mmmOk]J LF1$\"1(>8(4qp^:!#=7$$!1MLL[9cgJF1$\"1;$=!\\gn-FFC7$$!1nmmhN2-IF1$\"1( z$H#QbVS%FC7$$!1+++N&oz$GF1$\"1$*y>uQ+7rFC7$$!1nmm\")3DoEF1$\"1u2T:IzM 6!#<7$$!1+++:v2*\\#F1$\"1v_Hlj(ov\"FX7$$!1LLL8>1DBF1$\"1K)y;=(>tEFX7$$ !1nmmw))yr@F1$\"1J*=**p/Jx$FX7$$!1+++S(R#**>F1$\"1)*z_n2J2aFX7$$!1++++ @)f#=F1$\"1#>2>'=uJvFX7$$!1+++gi,f;F1$\"1VZ_Ar]25!#;7$$!1nmm\"G&R2:F1$ \"1luPQI&3G\"Fap7$$!1LLLtK5F8F1$\"1!p#*fH\\Pl\"Fap7$$!1MLL$yP2D\"F1$\" 1f8f!p1[#=Fap7$$!1MLL$HsV<\"F1$\"1Z\")eH$\\=+#Fap7$$!1nm;u5a&3\"F1$\"1 &4AHT;K@#Fap7$$!1-++]&)4n**Fap$\"1>`8/%owU#Fap7$$!1qmmTnt/GFap7$$!1)*****\\&y!pmFap$\"1W-2 cj'R>$Fap7$$!1******\\O3E]Fap$\"1R[@DF0;NFap7$$!1KLLL3z6LFap$\"1wD88W` wPFap7$$!1LLLeGmCDFap$\"18_?lgGkQFap7$$!1MLL$)[`P\"=bRFap7$$!11++]-6&)))FX$\"1%*[dclqtRFap7$$!1 SLLe4**RYFX$\"1,R`(fI^)RFap7$$!1Tnmmmr[RFC$\"1\"oq=q\"R*)RFap7$$\"1>L$ 3Fr)4=FX$\"1&=mIYp())RFap7$$\"16LL3Uh9SFX$\"1wKH(>4i)RFap7$$\"1/L$e9d$ >iFX$\"1!)=9OYr\")RFap7$$\"1(HLL3+TU)FX$\"12PdEBHvRFap7$$\"1GL$efeLG\" Fap$\"19!oZi/n&RFap7$$\"1FLL$=2Vs\"Fap$\"1>dR-SbIRFap7$$\"1hmmm7+#\\#F ap$\"1F\"Q>A`u'QFap7$$\"1'*****\\`pfKFap$\"1$zPDL/Iy$Fap7$$\"1HLLLm&z \"\\Fap$\"1`?Goi+NNFap7$$\"1(******z-6j'Fap$\"1F?6iE/-KFap7$$\"1#***** *4#32$)Fap$\"1fPs\")GGDGFap7$$\"1$****\\[9W#*eKEFap7$$\" 1%*****\\#y'G**Fap$\"125dFZ'pV#Fap7$$\"1****\\FJ*G3\"F1$\"1zNGu&z&>AFa p7$$\"1******H%=H<\"F1$\"1&oH4/o_+#Fap7$$\"1LLLo,\"QD\"F1$\"1=_l%H*z<= Fap7$$\"1mmm1>qM8F1$\"1)R**\\94rj\"Fap7$$\"1++++.W2:F1$\"1U%*z?hw!G\"F ap7$$\"1LLLep'Rm\"F1$\"1yN3\">QD***FX7$$\"1+++S>4N=F1$\"1J<6IZ=2uFX7$$ \"1mmm6s5'*>F1$\"1-!*QjXDTaFX7$$\"1+++lXTk@F1$\"1$yB)**[\"R$QFX7$$\"1m mmmd'*GBF1$\"1H:[qH-\\EFX7$$\"1+++DcB,DF1$\"1SLCoCUZ:nEF 1$\"1-!f%4Y7Q6FX7$$\"1LLL.a#o$GF1$\"10N=S#4^8(FC7$$\"1nmm^Q40IF1$\"1*= ^:5\"ekVFC7$$\"1+++!3:(fJF1$\"16hw7W\"*4FFC7$$\"1nmmc%GpL$F1$\"1b,=tr* Q_\"FC7$$\"1LLL8-V&\\$F1$\"1$zn%[cVn))F-7$$\"1+++XhUkOF1$\"1&))p3H7B%[ F-7$$\"1+++:oU6\"[:!#G7$F5$\"1Xr)= g#32:!#F7$F:$\"1n376#4_u\"!#E7$F?$\"12A,RB6E=!#D7$FE$\"1H+NdTk!o\"!#C7 $FJ$\"1())QBa-`=\"!#B7$FO$\"19y5')4ve!)F_bl7$FT$\"1hyU=zbB_!#A7$FZ$\"1 *)[^d.*4+$!#@7$Fin$\"1#yN=/5&3;!#?7$F^o$\"10V)**[MSQ'F^cl7$Fco$\"1NbnJ .$Hp#F-7$Fho$\"1YNe%yMO,\"FC7$F]p$\"1=+))\\)zbC$FC7$$!1LL$3x0Ke\"F1$\" 1/Vx\"*oB1`FC7$Fcp$\"1ywxB-2y%)FC7$$!1++]x#\\sT\"F1$\"1+bZ\"pokV\"FX7$ Fhp$\"1e:,nt,cBFX7$F]q$\"1y^wJ![F\\$FX7$Fbq$\"1_Z(Q$=ae]FX7$Fgq$\"1XHN ;))zdvFX7$F\\r$\"1?Xp:15%4\"Fap7$Far$\"1$3q7$)R&y9Fap7$Ffr$\"1eFvXHD\\ >Fap7$$!1ommTdc#Fap7$F[s$\"1nO*o1#4yKFap7$$!1*******4 \"eZeFap$\"1@g`c)[l-%Fap7$F`s$\"1t_M+)oT\"[Fap7$$!1LL$eW5vf%Fap$\"1&oK _r>\"G_Fap7$$!1mmmTs$*oTFap$\"1zc@dO5OcFap7$$!1****\\PSOSPFap$\"1G:x$) *[9.'Fap7$Fes$\"1`OAW#ysS'Fap7$Fjs$\"1\\y$3hwQ-(Fap7$F_t$\"1LWaM[L6vFa p7$Fdt$\"1'pT#f$>%3xFap7$Fit$\"1FR4Xm&Q&yFap7$$!1tm;/1binFX$\"1]@nx/?1 zFap7$F^u$\"14K4KLcWzFap7$$!12+]78V*y$ z'fyzFap7$Fhu$\"1#phU;?O(zFap7$F]v$\"1[%G:!y;`zFap7$Fbv$\"1tg4b(et\"zF ap7$Fgv$\"1LC*3(4SmyFap7$F\\w$\"106QdDI?xFap7$Faw$\"1\"RPy><#=vFap7$Ff w$\"19#)\\C>$p/(Fap7$F[x$\"1'HG`t)H^kFap7$$\"1imm\"*f#))3%Fap$\"1.K[\" Gp6r&Fap7$F`x$\"1iv<1d\")=\\Fap7$$\"1'****\\y#R*RcS\"FX7$F][l$\"1;T***>pdZ)FC7$$\"1mm;HOq&e\"F1$\" 15W)\\U&*GA&FC7$Fb[l$\"1O-^g3`SJFC7$Fg[l$\"1;B2=SF#[*F-7$F\\\\l$\"1?=K !*z=hFF-7$Fa\\l$\"11/$G)4l0oF^cl7$Ff\\l$\"1[c_Z.6^:F^cl7$F[]l$\"16c=hb \"p$HFjbl7$F`]l$\"17N$y8r^G&Ffbl7$Fe]l$\"1G`!f+()R;)F_bl7$Fj]l$\"1u75V F1V6F_bl7$F_^l$\"1jeqi6s)p\"F[bl7$Fd^l$\"1k<5]Ds)p\"Fgal7$Fi^l$\"1+@^% 3lv%>Fcal7$F^_l$\"1B$R:!G&=t\"F_al7$Fc_l$\"16@sz0/N:F[al7$Fh_lFe`l-F[` l6&F]`lF*F^`lF*-F$6$7io7$F($\"1#=*)p?^\"Rk!#Z7$F/$\"1_YP+TM!*H!#W7$F5$ \"1!=`OF7e+&!#U7$F:$\"1!*p9\\>HQ7!#R7$F?$\"1<>#4E*QQC!#P7$FE$\"1\\8`!* GQ(f$!#N7$FJ$\"1)RU*)Qif\"H!#L7$FO$\"1<^(Hq\\l<#!#J7$FT$\"1\"Hk6n=\"f9 Fg`l7$FZ$\"1$R'o0n4cuF[al7$Fin$\"1MsF[]EfKFcal7$F^o$\"1%y,esNlC(Fgal7$ Fco$\"1)*)Q'ea(y%=F_bl7$Fho$\"1r)y!ygpYOFfbl7$F]p$\"1](y;v&>,]Fjbl7$Fc p$\"1V#e4d[%QVF^cl7$Fhp$\"1aI(QP8eK%F-7$F]q$\"1P&QV6T!\\5FC7$Fbq$\"15u Dw=#RT#FC7$Fgq$\"1wO?3]QdfFC7$F\\r$\"1\"4,&\\jWp8FX7$Far$\"10n,g$Qkp#F X7$Ffr$\"1*G4T:=>-&FX7$F^fl$\"1#Rq;Hx$>$*FX7$F[s$\"1AYx3jQ<;Fap7$$!1** ***\\#)H$eiFap$\"17]!3[NR0#Fap7$Fffl$\"1**y/'\\2!pDFap7$$!1*****\\PKoV &Fap$\"1:(y7VB[;$Fap7$F`s$\"1rr6y*p+%QFap7$F^gl$\"1bS$))))4Ki%Fap7$Fcg l$\"1#pCZp2[Z&Fap7$Fhgl$\"13,]Oz(pP'Fap7$Fes$\"1YnOq\\.1tFap7$$!1LL$e% oA=HFap$\"1O\"*eirIe\")Fap7$Fjs$\"12d.aU)Q)*)Fap7$$!1LL$3())4J@Fap$\"1 ZCIV.0c(*Fap7$F_t$\"1P2:$o\"zW5F17$Fit$\"1U69Kb0b6F17$Fcu$\"1c^.oGu'> \"F17$Fgv$\"1*=9yk5#f6F17$Faw$\"1/'RA&o%p/\"F17$$\"1%****\\Aa\"3@Fap$ \"1k.'[#o%))z*Fap7$Ffw$\"170$QRq.0*Fap7$$\"1HLL3$[e(GFap$\"1Do%=C_*[#) Fap7$F[x$\"1\")>bJ'f%>uFap7$$\"1HL$3ngUn$Fap$\"1fMkJ1>>lFap7$Fd[m$\"1t E*4**Q-k&Fap7$$\"1'***\\78R.XFap$\"1N4ZjL*[![Fap7$F`x$\"1!etgkR/.%Fap7 $F\\\\m$\"17o_@O/2LFap7$Fa\\m$\"11v#yxc!pEFap7$Ff\\m$\"1LF_b l7$Fa\\l$\"1rFyuX-o$)Fgal7$Ff\\l$\"1ZS+mNR.IFcal7$F[]l$\"1'\\E.TvE5(F[ al7$F`]l$\"1KLli%G\")\\\"Fg`l7$Fe]l$\"11lwzK-TAFcdm7$Fj]l$\"1v$*yjvM(o #F_dm7$F_^l$\"1x(*RC*G]o$F[dm7$Fd^l$\"1tcvcyCs?Fgcm7$Fi^l$\"1<>gIw(\\e \"Fccm7$F^_l$\"1TK'zj#4WoF_cm7$Fc_l$\"1%yO4COQ$HF[cm7$Fh_lFebm-F[`l6&F ]`lF*F*F^`l-F$6$7io7$F($\"19w;\"H_Y5%!#r7$F/$\"1j)*)H(\\hhA!#m7$F5$\"1 +!)y9]AJ?!#i7$F:$\"1@jW30e_O!#e7$F?$\"1hiUbPWyV!#a7$FE$\"1&\\x*\\^KTJ! #]7$FJ$\"1`\"G<`!*=x(Fgbm7$FO$\"1(Qc47g1m\"!#V7$FT$\"17\"*=.ESJH!#S7$F Z$\"1h?*3\\4N>$Fgcm7$Fin$\"1\"*y6hUvNE!#M7$F^o$\"1S)o\")=&>Sl!#K7$Fco$ \"1rl&**4n12#Fg`l7$Fho$\"18D+OqdcTF_al7$F]p$\"1uVb=%p*oVFgal7$Fcp$\"1V 4/p'>U.#F_bl7$Fhp$\"1(***yie<87Fjbl7$Fbq$\"1l2*)))y;yDF^cl7$F\\r$\"1H( 3z2v?k&F-7$Far$\"1'eC0Nj;)=FC7$Ffr$\"1DJ)[_tVo&FC7$$!1-+]PL8jzFap$\"1! fn?(o1'***FC7$F^fl$\"1A?9\"fliq\"FX7$$!1LL$e9I/5(Fap$\"1K@GW,1FGFX7$F[ s$\"1!pTg4!oYXFX7$Fffl$\"1$[)eDK+N5Fap7$F`s$\"1lCVoa#\\6#Fap7$F^gl$\"1 v'QP9c;%HFap7$Fcgl$\"1VLNh_1tRFap7$Fhgl$\"1K`=:Xq5_Fap7$Fes$\"1fI5)\\\"F1 7$F^u$\"1hvl`&>&o:F17$Fcu$\"1l[_z+d&f\"F17$F]v$\"1CH]GeKv:F17$Fgv$\"1? !\\n?(p2:F17$F\\w$\"1MYDEDx)R\"F17$Faw$\"1&)f,-r'zD\"F17$F`[n$\"1HN!R8 ,$=6F17$Ffw$\"1=9\"QA'z4(*Fap7$Fh[n$\"1+*3(okBM#)Fap7$F[x$\"1x$y0h?-#o Fap7$F`\\n$\"1-McDv2>aFap7$Fd[m$\"1%*yWxf**)=%Fap7$Fh\\n$\"1j*zQz,.:$F ap7$F`x$\"1(*oP;'**[I#Fap7$Fa\\m$\"1!))4G#Ru26Fap7$Fex$\"1i;N`)eTt%FX7 $$\"1'****\\i(4]qFap$\"1Sfh<%>G*HFX7$F^]m$\"1Uo%[^#eR=FX7$$\"1$****\\F (3))yFap$\"1\"=r5a3%*4\"FX7$Fjx$\"1^(pkR;&)Q'FC7$F_y$\"1**=T]s=j?FC7$F dy$\"1H(=F!z'y*fF-7$F^z$\"1EW/#QC&\\EF^cl7$Fhz$\"1Q4y!=K>.\"Fjbl7$F][l $\"1-!)fY@,K?F_bl7$Fb[l$\"1-n_ua?IQFgal7$Fg[l$\"1^crz)pJ=$F_al7$F\\\\l $\"1wlHJbs)G#Fg`l7$Fa\\l$\"13_v(*ysY%)Fcen7$Ff\\l$\"17-M=.>zAF_en7$F[] l$\"1.d$*)fz$HHFgcm7$F`]l$\"1H_Zfb'=#F]cn7$Fh_lF gbn-F[`l6&F]`lF^`lF*F^`l-%+AXESLABELSG6$Q\"x6\"%!G-%%VIEWG6$;F(Fh_l%(D EFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curv e 1" "Curve 2" "Curve 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Int(b(50,x),x=-1/10..1/10); \nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*&*&-%$expG6 #,$*$)%\"xG\"\"#\"\"\"!%]7F1-%%sqrtG6#F0F1F1*$-F46#%#PiGF1!\"\"\"#D/F/ ;#F:\"#5#F1F?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+nU******!#5" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "6th description of " } {XPPEDIT 18 0 "delta(x)" "6#-%&deltaG6#%\"xG" }{TEXT -1 24 " as a limi t of functions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {XPPEDIT 18 0 "delta(x)" "6#-%&deltaG6#%\"xG" }{TEXT -1 59 " can be realised as the limit of the sequence of functions " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[n](x) = PIECEWISE([ n*cos(n*Pi*x/2)^2, abs(x) < 1/n],[0, 1/n <= abs(x)]);" "6#/-&%\"cG6#% \"nG6#%\"xG-%*PIECEWISEG6$7$*&F(\"\"\"*$-%$cosG6#**F(F0%#PiGF0F*F0\"\" #!\"\"F7F02-%$absG6#F**&F0F0F(F87$\"\"!1*&F0F0F(F8-F;6#F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 " Even though " }{XPPEDIT 18 0 "c[n](x);" "6#-&%\"cG6#%\"nG6#%\"xG" } {TEXT -1 81 " is defined by a piecewise formula, and is identically 0 \+ outside of the interval " }{XPPEDIT 18 0 "[-1/n,1/n]" "6#7$,$*&\"\"\"F &%\"nG!\"\"F(*&F&F&F'F(" }{TEXT -1 71 ", it is differentiable at all p oints, and the derivative is continuous." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Note that" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(cos(n*P i*x/2)^2,x = -1/n .. 1/n)=Int(1/2+cos(n*Pi*x)/2,x=-1/n..1/n)" "6#/-%$I ntG6$*$-%$cosG6#**%\"nG\"\"\"%#PiGF-%\"xGF-\"\"#!\"\"F0/F/;,$*&F-F-F,F 1F1*&F-F-F,F1-F%6$,&*&F-F-F0F1F-*&-F)6#*(F,F-F.F-F/F-F-F0F1F-/F/;,$*&F -F-F,F1F1*&F-F-F,F1" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = x/2+sin(n*Pi* x)/(2*n*Pi);" "6#/%!G,&*&%\"xG\"\"\"\"\"#!\"\"F(*&-%$sinG6#*(%\"nGF(%# PiGF(F'F(F(*(F)F(F0F(F1F(F*F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWI SE([1/n , ``],[ -1/n, ``])" "6#-%*PIECEWISEG6$7$*&\"\"\"F(%\"nG!\"\"%! G7$,$*&F(F(F)F*F*F+" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "`` = 1/n;" "6#/%!G*&\"\"\"F&%\"nG!\"\"" }{TEXT -1 28 ", for any positive integer " }{TEXT 270 1 "n" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 6 "Hence " }{XPPEDIT 18 0 "Int(c[n](x),x = -i nfinity .. infinity)=Int(n*cos(n*Pi*x/2)^2,x = -1/n .. 1/n)" "6#/-%$In tG6$-&%\"cG6#%\"nG6#%\"xG/F-;,$%)infinityG!\"\"F1-F%6$*&F+\"\"\"*$-%$c osG6#**F+F6%#PiGF6F-F6\"\"#F2F=F6/F-;,$*&F6F6F+F2F2*&F6F6F+F2" } {XPPEDIT 18 0 "``=1" "6#/%!G\"\"\"" }{TEXT -1 27 ", for any positive i nteger " }{TEXT 271 1 "n" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "Int(n*cos(n*Pi*x/2)^2 ,x=-1/n..1/n);\nassume(n_>0):\nsubs(n_=n,value(subs(n=n_,%)));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&%\"nG\"\"\")-%$cosG6#,$*(F' F(%#PiGF(%\"xGF(#F(\"\"#F2F(/F0;,$*&F(F(F'!\"\"F7F6" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 155 "c := (n,x) -> piecewise(abs(x)<1/n ,n*cos(n*Pi*x/2)^2,n*0.005):\nplot([c(1,x),c(2,x),c(3,x),c(4,x)],x=-2. .2,\n color=[red,green,blue,magenta]);" }}{PARA 13 "" 1 "" {GLPLOT2D 368 272 272 {PLOTDATA 2 "6(-%'CURVESG6$7gq7$$!\"#\"\"!$ \"1+++++++]!#=7$$!1LLL$Q6G\">!#:F+7$$!1nm;M!\\p$=F1F+7$$!1LLL))Qj^.\"F1F+7$$!1LL 37&)=@5F1F+7$$!1+Dc^jz:5F1F+7$$!1n;/\">//,\"F1F+7$$!1]7y5\"3x+\"F1F+7$ $!1L3_I?,05F1F+7$$!1'***!#;$\"1qEMu\\OlN!#A7$$ !1,++vE\"z))*Fho$\"1y0w[fh*4$!#>7$$!1,++]miz(*Fho$\"1I4rA.!y>\"F-7$$!1 +++D1Mr'*Fho$\"1&41mKXGm#F-7$$!1,+++Y0j&*Fho$\"1Y:*)y7S.ZF-7$$!1+++]D[ Y$*Fho$\"10!3)eP4]5!#<7$$!1++++0\"*H\"*Fho$\"1[cAt\"ej&=Ffq7$$!1++++f \\7()Fho$\"1E2:y#oY.%Ffq7$$!1++++83&H)Fho$\"1Tihg*zA+(Ffq7$$!1nm;a)Gg \"zFho$\"1od8!GVQ.\"Fho7$$!1LLL3k(p`(Fho$\"1$Q\"y)4NOU\"Fho7$$!1++](QY i3(Fho$\"15'[@*Qc_>Fho7$$!1nmmmj^NmFho$\"15NZHV\\UDFho7$$!1omm;*)o`iFh o$\"1x@Rq1B\"3$Fho7$$!1ommm9'=(eFho$\"1_=r5LaZOFho7$$!1ML$3P0xU&Fho$\" 1`V5%y\"=IVFho7$$!1,++v#\\N)\\Fho$\"1()*pLiSe-&Fho7$$!1NL$3(eR!f%Fho$ \"1N!3[YI;k&Fho7$$!1pmmmCC(>%Fho$\"1Q)o\"4WkZiFho7$$!1ML$3(3*ew$Fho$\" 1cB$)f@L!*oFho7$$!1*****\\FRXL$Fho$\"1G:X=$f$)\\(Fho7$$!1******\\0zBHF ho$\"1\")f)4x@\\.)Fho7$$!1+++D=/8DFho$\"1BTKX%=5_)Fho7$$!1LL$3ioW3#Fho $\"11QhT3ol*)Fho7$$!1mmm;a*el\"Fho$\"11-GJ>cQ$*Fho7$$!1nm;H9Li7Fho$\"1 \"=vhP]>h*Fho7$$!1pmm;Wn(o)Ffq$\"17_]zV#\\\")*Fho7$$!1OL$3x9^c'Ffq$\"1 %[A/mHS*)*Fho7$$!1.++D^bUWFfq$\"1$))))yj\"Q^**Fho7$$!1OL3-`F\"Q$Ffq$\" 1T#RQm;=(**Fho7$$!1qm;za**>BFfq$\"1*3!3!RDn)**Fho7$$!1.+Dccre7Ffq$\"18 f=X74'***Fho7$$!1qLLL$eV(>F-$\"1uBx=Q!*****Fho7$$\"1$fmTNc$\\!*F-$\"14 GMg&zz***Fho7$$\"1cm;/rI2?Ffq$\"1\">t)R91!***Fho7$$\"1_m\"Hdy'4JFfq$\" 1>]A`*eh(**Fho7$$\"1[mmT+07UFfq$\"19H0t()Gc**Fho7$$\"1Tm;zHz;kFfq$\"1t #p\"yxu)*)*Fho7$$\"1Mmm;f`@')Ffq$\"1yCB)R9x\")*Fho7$$\"1JLLL1+Y7Fho$\" 1B2'f5)z@'*Fho7$$\"1)****\\nZ)H;Fho$\"1fVkoUve$*Fho7$$\"1JL$e*HTW?Fho$ \"1YB$fW'o.!*Fho7$$\"1lmm;$y*eCFho$\"1td&fo-3e)Fho7$$\"1KLLe[E()GFho$ \"1;%\\!3sJ!3)Fho7$$\"1*******R^bJ$Fho$\"1+mb40:CvFho7$$\"1(****\\AYXt $Fho$\"1$GUEX@e$pFho7$$\"1'*****\\5a`TFho$\"1)>dtI**RJ'Fho7$$\"1'***\\ (3S*eXFho$\"1hU-T/g!p&Fho7$$\"1(****\\7RV'\\Fho$\"1-Bp*z9g0&Fho7$$\"1( ***\\PcY9aFho$\"1OG\"\\)pz]VFho7$$\"1'*****\\@fkeFho$\"1)*4P&yR&eOFho7 $$\"1jmmT30piFho$\"1H9!\\=q*eIFho7$$\"1JLLL&4Nn'Fho$\"1m;a$4(p!\\#Fho7 $$\"1lmm;bN0rFho$\"1N!\\=&>#)G>Fho7$$\"1*******\\,s`(Fho$\"1j\"G[,)QB9 Fho7$$\"1KL$e9=&GzFho$\"1>9bez#>-\"Fho7$$\"1lmm\"zM)>$)Fho$\"1VbFWh80o Ffq7$$\"1KL$eCZwu)Fho$\"1r,7 F-7$$\"1i;a8:!*z)*Fho$\"1M>cT*p%eNFap7$$\"1HLLeg`!)**Fho$\"1$*o>%>(fZ$ *!#@7$$\"1m\"H#3Mo+5F1F+7$$\"1+]i5KJ.5F1F+7$$\"1L3-8I%f+\"F1F+7$$\"1mm T:Gd35F1F+7$$\"1L$3-UKQ,\"F1F+7$$\"1+++D?4>5F1F+7$$\"1LLeM7hH5F1F+7$$ \"1nm;W/8S5F1F+7$$\"1LLLj)o61\"F1F+7$$\"1++]#G2A3\"F1F+7$$\"1LLL$)G[k6 F1F+7$$\"1++]7yh]7F1F+7$$\"1nmm')fdL8F1F+7$$\"1nmm,FT=9F1F+7$$\"1LL$e# pa-:F1F+7$$\"1+++Sv&)z:F1F+7$$\"1LLLGUYo;F1F+7$$\"1nmm1^rZF1F+7$$\"\"#F*F+-%'COLOURG6&%$RGBG$\"*++++\" !\")F*F*-F$6$7_q7$F($\"1+++++++5Ffq7$F/Fihl7$F3Fihl7$F6Fihl7$F9Fihl7$F HD!>M&!#?7$$!1M$eR#4E&)[Fho$\"1*\\w>[_vf#F-7$$! 1om\"Hdspy%Fho$\"1z(z%HCUW*)F-7$$!1,](=A%o)o%Fho$\"1q;u`I)p!>Ffq7$F_u$ \"1*>epAQR%Fho$\"1d'Q3\\9g;(Ffq7$Fdu$\"13UQdKHX7Fho 7$$!1,+vomc\")RFho$\"1_(fjX[%y>Fho7$Fiu$\"1mrfRXoeGFho7$$!1mm\"H2:-b$F ho$\"1_))**[<')pQFho7$F^v$\"1p[FF%RM*\\Fho7$$!1****\\7\\;HJFho$\"1QDU! )=+\\hFho7$Fcv$\"1\\P![7+'otFho7$$!1++](=;%=FFho$\"1hqj)Fho7$Fhv$ \"1s,&Qrc!=**Fho7$$!1mm\"HAb()H#Fho$\"1OuVW\"4h7\"F17$F]w$\"1CM+j*H\"e 7F17$$!1++v=?=q=Fho$\"17Gfn%yaQ\"F17$Fbw$\"11V_q&\\e]\"F17$Fgw$\"1\">% y,pg,F17$Fjy$\"1lzvbI#***>F 17$F_z$\"1h$[[(RQ)*>F17$Fdz$\"1Xn*Q0d?*>F17$Fiz$\"1B4AME(4)>F17$F^[l$ \"1H7:sQ=l>F17$Fc[l$\"1s%HNQ=)>>F17$Fh[l$\"1A\\$=wHo&=F17$F]\\l$\"1q4e =9))35**)>:F17$$\"1lmTN.8P=Fho$\"1.*zx,cXS\"F17$Fg\\ l$\"1!R(e7/O#G\"F17$$\"1)**\\il&p^AFho$\"1\\P?\"3#Qb6F17$F\\]l$\"1X)f+ $=xD5F17$$\"1)***\\(e@Jn#Fho$\"1Ro;*y)Q9*)Fho7$Fa]l$\"1!GkK9`I&FhoFihl7$F__lFihl7$$\"1(**\\ P*)G&RcFhoFihl7$Fd_lFihl7$F^`lFihl7$Fh`lFihl7$FbalFihl7$F\\blFihl7$Fjc lFihl7$F[flFihl7$F^flFihl7$FaflFihl7$FdflFihl7$FgflFihl7$FjflFihl7$F]g lFihl7$F`glFihl7$FcglFihl7$FfglFihl7$FiglFihl7$F\\hlFihl-F_hl6&FahlF*F bhlF*-F$6$7ip7$F($\"1+++++++:Ffq7$F/F`im7$F3F`im7$F6F`im7$F9F`im7$F$G$Fho$\"1hu>ApOu;F-7$$!1***\\P4_=B$Fho$\"1S%f!QgdboF-7 $$!1**\\7.&30=$Fho$\"1x7\")=mC`:Ffq7$Fd^m$\"1y7jVBZoFFfq7$$!1***\\7txk -$Fho$\"1m#oT#oMHiFfq7$Fcv$\"1<.SD!zN5\"Fho7$F\\_m$\"1ng:+vK\\CFho7$Fh v$\"18_#3PIQE%Fho7$Fd_m$\"1hy%=y4Le'Fho7$F]w$\"1ENtoP#\\C*Fho7$$!1m;z> `Kx>Fho$\"1+$>4pkq1\"F17$F\\`m$\"1Xl.&*y/97F17$$!1L$3xrQIw\"Fho$\"1X^1 vW%RO\"F17$Fbw$\"1bYRMsA::F17$$!1mm\"HU8\"f9Fho$\"1C&\\)oPb\"z\"F17$Fg w$\"1pQ\"fh!)y0#F17$$!1nmTN%\\b1\"Fho$\"1LN5>L20BF17$F\\x$\"1m2C$o`Y_# F17$Fax$\"1\"*ehb$3>s#F17$Ffx$\"1@(oD!oUqGF17$F`y$\"13&GfJ&GkHF17$Fjy$ \"1gfM<.u**HF17$$\"17mT5!*\\PNF-$\"1yPQ.k;**HF17$F_z$\"12np^xa%*HF17$$ \"1e;zp87c9Ffq$\"1PiOBo*e)HF17$Fdz$\"1]1_epBtHF17$Fiz$\"1,3-ox.OHF17$F ^[l$\"1_!pW$>(\\H6[Fho7$Fhcm$\"1'[\\ @0K8\"GFho7$Fa]l$\"1q\"ex_shI\"Fho7$F`dm$\"1wFya*)=pNFfq7$Ff]l$\"1L<\" eu]k5#Fap7$$\"1*\\ildQh\">#Fho$\"1FZs8#3J[\"Fho7$F]w$\"1bf+7JClEFho7$F^]n$\"1OvLlz9hTFho7$F \\`m$\"1\"43\"ffuVfFho7$Ff]n$\"1I7L`4x!)zFho7$Fbw$\"1*\\)*>:NN-\"F17$$ !1m;z>W]d:Fho$\"1I-4!pZiC\"F17$F^^n$\"1(yf\\Qo/[\"F17$$!1m;/ECsg8Fho$ \"1'\\=l>@Es\"F17$Fgw$\"1[[_y*3!p>F17$$!1n;HK/%R;\"Fho$\"1f&ef#*pe@#F1 7$Ff^n$\"1'R$f4]VfCF17$$!1omT&Q%er'*Ffq$\"14z'3_&)fp#F17$F\\x$\"1i\"p% [)4>#HF17$$!1-+v$f%REwFfq$\"1y![_sB(\\JF17$Fax$\"1eExZ)=rN$F17$$!1pm\" z%\\$Q]&Ffq$\"1lr$)R>TSNF17$Ffx$\"1T!)=qxM'p$F17$F`y$\"1K?])=0c\"RF17$ Fjy$\"1#QX(pWQ**RF17$Fdz$\"1`^u#z3n$RF17$F^[l$\"1jLCG'>js$F17$$\"1XmT5 :U9`Ffq$\"1'yC!Q[LqNF17$Fc[l$\"1'3:ZVjUQ$F17$$\"1Qm\"zWk\">vFfq$\"10^i F/nrJF17$Fh[l$\"1`V1j*Gm$HF17$$\"1-L$3_`6e*Ffq$\"1*\\*3JtCM\"Fho$\"1!\\DtR(Qp0:;De)Fho7$Fhbm$\"1Mt(3'GiYlFho7$$\"1)*\\il;xS>Fho$\"1uf'4!zgQZF ho7$Fg\\l$\"1X3'Q5'4*=$Fho7$F`cm$\"1TMojkVd'*Ffq7$F\\]l$\"1NT`)*)[nl#F -7$$\"1)*\\PMT^7DFhoF\\gn7$$\"1JL3_*\\gc#FhoF\\gn7$$\"1l;zpde>EFhoF\\g n7$FhcmF\\gn7$$\"1lm\"HA$>!y#FhoF\\gn7$Fa]lF\\gn7$F`dmF\\gn7$Ff]lF\\gn 7$F`^lF\\gn7$Fj^lF\\gn7$Fd_lF\\gn7$F^`lF\\gn7$Fh`lF\\gn7$FbalF\\gn7$F \\blF\\gn7$FjclF\\gn7$F[flF\\gn7$F^flF\\gn7$FaflF\\gn7$FdflF\\gn7$Fgfl F\\gn7$FjflF\\gn7$F]glF\\gn7$F`glF\\gn7$FcglF\\gn7$FfglF\\gn7$FiglF\\g n7$F\\hlF\\gn-F_hl6&FahlFbhlF*Fbhl-%+AXESLABELSG6$Q\"x6\"%!G-%%VIEWG6$ ;F(F\\hl%(DEFAULTG" 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" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Int('c( 100,x)',x=-1/100..1/100);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#-%$IntG6$-%\"cG6$\"$+\"%\"xG/F*;#!\"\"F)#\"\"\"F)" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"\"\"\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 19 "7th description of " }{XPPEDIT 18 0 "delta(x)" "6#-%&de ltaG6#%\"xG" }{TEXT -1 24 " as a limit of functions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {XPPEDIT 18 0 "delta(x )" "6#-%&deltaG6#%\"xG" }{TEXT -1 59 " can be realised as the limit of the sequence of functions " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "v[n](x) = PIECEWISE([``(n/A)*exp(1/(n^2*x^2-1)), abs(x) < 1/n],[0, 1/n <= abs(x)]);" "6#/-&%\"vG6#%\"nG6#%\"xG-%*PIECEWISEG6$ 7$*&-%!G6#*&F(\"\"\"%\"AG!\"\"F4-%$expG6#*&F4F4,&*&F(\"\"#F*F=F4F4F6F6 F42-%$absG6#F**&F4F4F(F67$\"\"!1*&F4F4F(F6-F@6#F*" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 7 " where " }{XPPEDIT 18 0 "A =`` " "6#/%\"AG%!G" }{XPPEDIT 18 0 "Int(exp(1/(x^2-1 )),x = -1 .. 1) = Limit(``,epsilon = 0);" "6#/-%$IntG6$-%$expG6#*&\"\" \"F+,&*$%\"xG\"\"#F+F+!\"\"F0/F.;,$F+F0F+-%&LimitG6$%!G/%(epsilonG\"\" !" }{XPPEDIT 18 0 "Int( exp(1/(x^2-1)),x=-1+epsilon..1-epsilon)" "6#-% $IntG6$-%$expG6#*&\"\"\"F*,&*$%\"xG\"\"#F*F*!\"\"F//F-;,&F*F/%(epsilon GF*,&F*F*F3F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 12 "Even th ough " }{XPPEDIT 18 0 "v[n](x)" "6#-&%\"vG6#%\"nG6#%\"xG" }{TEXT -1 81 " is defined by a piecewise formula, and is identically 0 outside o f the interval " }{XPPEDIT 18 0 "[-1/n, 1/n]" "6#7$,$*&\"\"\"F&%\"nG! \"\"F(*&F&F&F'F(" }{TEXT -1 114 ", it has derivatives of all orders at all points, that is, the graphs of all of its derivatives are smooth \+ curves." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "Int(exp(1/(x^2-1)),x=-1..1);\nA := evalf(evalf(%,35), 30);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%$expG6#*&\"\"\"F*,& *$)%\"xG\"\"#F*F*F*!\"\"F0/F.;F0F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"AG$\"?r6#*[I#yVz!oh\"Q*RW!#I" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Letting " }{XPPEDIT 18 0 "u = n*x" "6#/%\"uG*&%\"nG\"\"\"% \"xGF'" }{TEXT -1 11 " shows that" }}{PARA 256 "" 0 "" {TEXT -1 2 " \+ " }{XPPEDIT 18 0 "Int(``(n/A)*exp(1/(n^2*x^2-1)),x = -1/n .. 1/n) = 1/ A;" "6#/-%$IntG6$*&-%!G6#*&%\"nG\"\"\"%\"AG!\"\"F--%$expG6#*&F-F-,&*&F ,\"\"#%\"xGF6F-F-F/F/F-/F7;,$*&F-F-F,F/F/*&F-F-F,F/*&F-F-F.F/" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(exp(1/(x^2-1)),x = -1 .. 1)" "6#-%$IntG6$ -%$expG6#*&\"\"\"F*,&*$%\"xG\"\"#F*F*!\"\"F//F-;,$F*F/F*" }{TEXT -1 6 " = 1, " }}{PARA 0 "" 0 "" {TEXT -1 25 "for any positive integer " } {TEXT 272 1 "n" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "v := (n,x) -> piecewise(abs (x)<1/n,n/A*exp(1/(n^2*x^2-1)),n*0.005):\nplot([v(1,x),v(2,x),v(3,x),v (4,x)],x=-2..2,\n color=[red,green,blue,magenta]);" }} {PARA 13 "" 1 "" {GLPLOT2D 360 270 270 {PLOTDATA 2 "6(-%'CURVESG6$7[q7 $$!\"#\"\"!$\"35+++++++]!#?7$$!3MLLL$Q6G\">!#.\"F1F+7$$!3]LL37&)=@5F1F+7$$!39+Dc^jz:5F1F+7$$!3 zm;/\">//,\"F1F+7$$!3+]7y5\"3x+\"F1F+7$$!3VL3_I?,05F1F+7$$!3(oTg-&fJ-5 F1F+7$$!3w++++()>'***!#=$F*F*7$$!3k+++]miz(*Fho$\"3O.!=Lqv.X#!#F7$$!3_ ++++Y0j&*Fho$\"3/1(oH)f]q=!#A7$$!3-,++v&oZX*Fho$\"3%[JDX]yF\"=!#@7$$!3 Q+++]D[Y$*Fho$\"3[`aWZ`[s#)F[q7$$!3x*****\\_'>Q#*Fho$\"3kctlCD$3X#F-7$ $!3E++++0\"*H\"*Fho$\"3,T9&e_)))QbF-7$$!3=++++f\\7()Fho$\"3#f)3ghi;[N! #>7$$!35++++83&H)Fho$\"3s7GX9!))f7*F`r7$$!3!ommT&)Gg\"zFho$\"3+Qb7gUzY :Fho7$$!3\\LLL3k(p`(Fho$\"3D!e4bn*=CAFho7$$!3M++](QYi3(Fho$\"3=Qw'fHM> -$Fho7$$!3Anmmmj^NmFho$\"3#4o2o)[%Hx$Fho7$$!3gnmm;*)o`iFho$\"31!\\H%)) *z!fVFho7$$!3)zmmmYh=(eFho$\"3KEG00ZX&*[Fho7$$!3\\ML$3P0xU&Fho$\"3y*\\ ZV>vpX&Fho7$$!3+,++v#\\N)\\Fho$\"3IW;G3qFafFho7$$!3wML$3(eR!f%Fho$\"3D V4P.$)HWjFho7$$!3commmCC(>%Fho$\"3#RXY5300p'Fho7$$!39*****\\FRXL$Fho$ \"3geb8o,N6tFho7$$!3t*****\\#=/8DFho$\"3'3&f^?:bXxFho7$$!3=mmm;a*el\"F ho$\"3U&e0&GENb!)Fho7$$!3_mm;H9Li7Fho$\"3%R\")[90+E:)Fho7$$!3komm;Wn(o )F`r$\"3=w9J.]\"HA)Fho7$$!3$G++]7bDW%F`r$\"3%\\(Ruq$>$p#)Fho7$$!3IqLLL $eV(>F-$\"3kB=5+hl&G)Fho7$$\"3qbm;/rI2?F`r$\"3a#*)*[)=\\BG)Fho7$$\"3V[ mmT+07UF`r$\"3%4x&H3a(4F)Fho7$$\"3:Tm;zHz;kF`r$\"33dp8]:]^#)Fho7$$\"3) Qjmm\"f`@')F`r$\"3r_bs%oqQA)Fho7$$\"3mILLL1+Y7Fho$\"3_qp7Hz/c\")Fho7$$ \"3%z****\\nZ)H;Fho$\"3]vwP,!QE1)Fho7$$\"3ckmm;$y*eCFho$\"3gx\\g!*=Cpx Fho7$$\"3f)******R^bJ$Fho$\"3s#4]aUHIK(Fho7$$\"3'e*****\\5a`TFho$\"3`a GN#zg]kL%Fho7$$\"3_I LLL&4Nn'Fho$\"3\")[q82n)>r$Fho7$$\"3([mmm^b`5(Fho$\"3E0s[z6()))HFho7$$ \"3A*******\\,s`(Fho$\"3)HVK2=&yBAFho7$$\"3.KL$e9=&GzFho$\"3q\\:06l([_ \"Fho7$$\"3%[mm;zM)>$)Fho$\"3H*RQpW4Lu)F`r7$$\"33KL$eCZwu)Fho$\"3U&ou! H\\L$=$F`r7$$\"3M*******pfa<*Fho$\"3/vhBIL$[.%F-7$$\"35m;zWU4w#*Fho$\" 3v]7?sF/R7F[q7$$\"3Ckm;zy*zd*Fho$\"3k<*ou`svC\"Fep7$$\"3o'**\\(opEz (*Fho$\"3)Q%z_$Q_Ea#F_p7$$\"39HLLeg`!)**Fho$\"3OyB,SPD(y%!$H\"7$$\"3Cm \"H#3Mo+5F1F+7$$\"3c**\\i5KJ.5F1F+7$$\"3*G$3-8I%f+\"F1F+7$$\"3@mmT:Gd3 5F1F+7$$\"3'GL3-UKQ,\"F1F+7$$\"3^*****\\-#4>5F1F+7$$\"3\"GL$eM7hH5F1F+ 7$$\"3Lmm;W/8S5F1F+7$$\"3;LLLj)o61\"F1F+7$$\"3w****\\#G2A3\"F1F+7$$\"3 ;LLL$)G[k6F1F+7$$\"3#)****\\7yh]7F1F+7$$\"3xmmm')fdL8F1F+7$$\"3bmmm,FT =9F1F+7$$\"3FLL$e#pa-:F1F+7$$\"3!*******Rv&)z:F1F+7$$\"3ILLLGUYo;F1F+7 $$\"3_mmm1^rZF1F+7$$\"\"#F*F+- %'COLOURG6&%$RGBG$\"*++++\"!\")FioFio-F$6$7eq7$F($\"3-+++++++5F`r7$F/F ]el7$F3F]el7$F6F]el7$F9F]el7$FQR%Fho$\"3%)3EAhjB%e&F`r7$Fdu$\"3@@'[G\"[QC:Fho7$$!3?,+vomc\")RFh o$\"3#Rn6P^d(GHFho7$$!3%QLL3(3*ew$Fho$\"3/!z8#Q![qY%Fho7$$!3]mm\"H2:-b $Fho$\"3%fa_f)G![*fFho7$Fiu$\"3!o\">6\"[R#QuFho7$$!3H****\\7\\;HJFho$ \"3xk$R7;/Xq)Fho7$$!3V******\\0zBHFho$\"3S[B9L*Qd&)*Fho7$$!3e****\\(=; %=FFho$\"3iSLR$>>#*3\"F17$F^v$\"3'>N5[@k==\"F17$$!3%HLL3ioW3#Fho$\"3Pw -\\[XwU8F17$Fcv$\"3#po/rdm]Y\"F17$Fhv$\"3=Bzn5n2[:F17$F]w$\"3#f&H$4]Xj g\"F17$$!3sNL$3x9^c'F`r$\"39nVbe/KG;F17$Fbw$\"3lN%>4g.Sk\"F17$$!3QOL3- `F\"Q$F`r$\"3!3Ppt)=a\\;F17$$!3$*pm;za**>BF`r$\"3%4@M\"*4mNl\"F17$$!3[ .+Dccre7F`r$\"3!R(*4C9(3c;F17$Fgw$\"3_Q>\"3%=6d;F17$$\"3O$fmTNc$\\!*F- $\"3C#RPJx%fc;F17$F\\x$\"3;o4O'pkWl\"F17$$\"31_m\"Hdy'4JF`r$\"3/$3J:R: 2l\"F17$Fax$\"35.1^**eLX;F17$Ffx$\"3c/h%)y%='H;F17$F[y$\"3-srS8u72;F17 $F`y$\"3wj`kO!o4b\"F17$Fey$\"3qFtmjLQr9F17$$\"3DJL$e*HTW?Fho$\"3%pu*)) *oOdN\"F17$Fjy$\"38?rVLa]/7F17$$\"31)***\\(e@Jn#Fho$\"3'>rIFbv06\"F17$ $\"3cJLLe[E()GFho$\"3Y/@N$eO[+\"F17$$\"33lm;H\"395$Fho$\"3W`![\"z@)o') )Fho7$F_z$\"3e(>m$yC**fvFho7$$\"3\"z***\\7)[]_$Fho$\"3+*\\l%HzZohFho7$ $\"3A(****\\AYXt$Fho$\"3C=bbQX5#p%Fho7$$\"3`'***\\PO/WRFho$\"3-$y8k'H( >>$Fho7$Fdz$\"3qMR+^\\'zy\"Fho7$$\"35'**\\(o0CcVFho$\"3!f=td,Zm4(F`r7$ Fiz$\"3#fDD(\\5*y>\"F`r7$$\"3T'\\(=n\\h4YFho$\"3Uf4_;RXZdF-7$$\"3['*\\ (o%)*GgYFho$\"3e>7$Hh)39AF-7$$\"3a'\\ilsk4r%Fho$\"36oDbC*e95'F[q7$$\"3 g'**\\igR;w%Fho$\"3RzGg1FC9(*Fep7$$\"3u'*\\il$*)H'[Fho$\"3M!p4!yu5gT!# D7$F^[l$\"3Y=mQoXra7!#Z7$$\"3cY7.K?Z#*\\Fho$\"3WZ^,$\\0(z?!$h\"7$$\"3F '\\i!R\\g?]FhoF]el7$$\"35ZP4Yyt[]FhoF]el7$$\"3!o*\\7`2(o2&FhoF]el7$$\" 3L(\\(=nl8L^FhoF]el7$$\"3w'**\\7Q-%*=&FhoF]el7$$\"3q'*\\P4S$>I&FhoF]el 7$Fc[lF]el7$$\"3b'**\\P*)G&RcFhoF]el7$Fh[lF]el7$Fb\\lF]el7$F\\]lF]el7$ Ff]lF]el7$F`^lF]el7$F^`lF]el7$F_blF]el7$FbblF]el7$FeblF]el7$FhblF]el7$ F[clF]el7$F^clF]el7$FaclF]el7$FdclF]el7$FgclF]el7$FjclF]el7$F]dlF]el7$ F`dlF]el-Fcdl6&FedlFioFfdlFio-F$6$7eq7$F($\"3%**************\\\"F`r7$F /Ffhm7$F3Ffhm7$F6Ffhm7$F9Ffhm7$F$G$Fho$\"3A]F[?CO@>!#J7$$!37*\\i!*)Q_dKFho$\"3G__r 9'oe[\"Fbhl7$$!3\\***\\P4_=B$Fho$\"3#H?E@\\'ziQ!#C7$$!37**\\7.&30=$Fho $\"3qs:OB)o6g*Fep7$F[[m$\"3%)4s1i9o([\"F-7$$!33***\\7txk-$Fho$\"35Qrqp _cvAF`r7$F`[m$\"32\"eqV(fSW))F`r7$$!3y***\\(oL5@GFho$\"3QxteF>!4*>Fho7 $Fe[m$\"3(QlV?9m@T$Fho7$$!3Q***\\i+Hdh#Fho$\"3J@x3$zG]+&Fho7$F^v$\"3S2 1^8I)4m'Fho7$$!3/L$eR_)*eS#Fho$\"3'4PgntA&y$)Fho7$$!3Mmm\"HAb()H#Fho$ \"3K0@\\ywn.5F17$$!3k**\\(=#>h\">#Fho$\"3_BaX-;!3;\"F17$F]\\m$\"3X\\() Hd)[yI\"F17$$!3Em;z>`Kx>Fho$\"3$>Iqs\"**HW9F17$$!3c***\\(=?=q=Fho$\"3S 8AwWJ9q:F17$$!3'GL3xrQIw\"Fho$\"3w1N'=Dcco\"F17$Fcv$\"3E*e*fH1F\"z\"F1 7$$!3Mmm\"HU8\"f9Fho$\"3%fC#=B_9h>F17$Fhv$\"3')RdyV?^-@F17$$!3ommTN%\\ b1\"Fho$\"3lL>*[k*H=AF17$F]w$\"3'=R\"\\@`*4J#F17$F[]m$\"3Qe)zbF!R(Q#F1 7$Fbw$\"37*f:vHf6W#F17$Fc]m$\"3[5k3`s**fCF17$Fh]m$\"3I6K$\\>OOZ#F17$F] ^m$\"39'R*H^&f@[#F17$Fgw$\"3)pTTCJ>c[#F17$Fe^m$\"3q^XpMQ(Q[#F17$F\\x$ \"3U%3%*[5wmZ#F17$F]_m$\"3KhtDz(yRY#F17$Fax$\"3WY8cE\")pXCF17$Ffx$\"3y ?+A%Hk=R#F17$F[y$\"36[=f)pMPJ#F17$$\"3-(***\\7r2a5Fho$\"3eB+$)ofJCAF17 $F`y$\"3*\\5OS=dI6#F17$$\"3Ikm;aT#zV\"Fho$\"3iFTo,>ox>F17$Fey$\"3u\\#p ^)o]:=F17$$\"3gkmTN.8P=Fho$\"3#f*)Q\")fgog\"F17$Fa`m$\"3uGdi#R5,O\"F17 $$\"3ek;/EV0[@Fho$\"3s!GCbIT=A\"F17$$\"3!z**\\il&p^AFho$\"3#oq=`knQ2\" F17$$\"3CJ$ek)pLbBFho$\"3Q!R7*\\Abq\"*Fho7$Fjy$\"3jatBTwdKvFho7$$\"3IJ L3_*\\gc#Fho$\"3n&*p#eG)\\.eFho7$Fi`m$\"3.8R?Mh&45%Fho7$$\"3#[m;HA$>!y #Fho$\"3tT[S'*R)z_#Fho7$F^am$\"3vA!or5#GK7Fho7$$\"3K)**\\P\\OV*HFho$\" 3H58L)*>H.QF`r7$Fcam$\"3AK)Q9?xu%RF-7$$\"3%=L$ek(z%3KFho$\"3_v(3=2J0N) !#B7$F_z$\"30^x)o(y#e.\"!#d7$$\"3^)\\ildQ2osJR\"F`r7$F_ ^n$\"3aN'GA@$[)>\"Fho7$F]\\m$\"3#)\\jZ6#zqQ$Fho7$Fg^n$\"3%zV/9QwWB'Fho 7$F\\_n$\"3gPB%HAW2I*Fho7$Fa_n$\"3L*>:1s!GK7F17$Fcv$\"3xFW]d:Fho$\"3#RTi`W;vv\"F17$Fi_n$\"3)=F17$$!3Wm;/ECsg8Fh o$\"3=fK0\"HWb<#F17$Fhv$\"3s_*Rpx3QN#F17$Fa`n$\"3q$Ge1ByWl#F17$F]w$\"3 +$*Rc(*e%*))GF17$Fbw$\"3<)=jGBZz?$F17$Fgw$\"3JvQUV'oSJ$F17$Fax$\"3C&\\ l0\"z%)=KF17$F[y$\"3O5R=3[y&*GF17$F]cn$\"3%zLx#)*=()pEF17$F`y$\"3[nlxS ._\"Q#F17$$\"3[(**\\PRi>M\"Fho$\"3wY_t&Hg5@#F17$Fecn$\"3v)z<3*fc@?F17$ $\"37JLe9f)Q`\"Fho$\"3S\")=E*zi@\"=F17$Fey$\"3+Edg+]a#e\"F17$$\"3FJ$3_ +*[LFho$\" 39V#eP5MEF(Fho7$Fa`m$\"3dn`dD=&>S%Fho7$Fedn$\"3Q%))p++#Hu>Fho7$Fjdn$\" 3#\\Q&*e1k&4XF`r7$$\"3ckTN@j^.BFho$\"3M_\"*e@E\"))>\"F`r7$F_en$\"3Z_f# )ya$4B\"F-7$$\"3*y\\i:ldrS#Fho$\"3aIx2%zrx#**Fcgn7$Fjy$\"3W&H7IxH\")3% Fj\\o7$$\"3CJ3_Diu&[#Fho$\"3%3#\\Bb#3Gv&!#b7$$\"3$z*\\PMT^7DFhoFgjn7$$ \"3ik\"HK/#GRDFhoFgjn7$FgenFgjn7$$\"3ok;zpde>EFhoFgjn7$Fi`mFgjn7$F_fnF gjn7$F^amFgjn7$FcamFgjn7$F_zFgjn7$FdzFgjn7$F^[lFgjn7$Fh[lFgjn7$Fb\\lFg jn7$F\\]lFgjn7$Ff]lFgjn7$F`^lFgjn7$F^`lFgjn7$F_blFgjn7$FbblFgjn7$FeblF gjn7$FhblFgjn7$F[clFgjn7$F^clFgjn7$FaclFgjn7$FdclFgjn7$FgclFgjn7$FjclF gjn7$F]dlFgjn7$F`dlFgjn-Fcdl6&FedlFfdlFioFfdl-%+AXESLABELSG6$Q\"x6\"Q! Fjgo-%%VIEWG6$;F(F`dl%(DEFAULTG" 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" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "Int('v(100,x)',x=-1/100..1/100);\nevalf(%,20);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%$IntG6$-%\"vG6$\"$+\"%\"xG/F*;#!\"\"F)#\"\"\"F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5+++++++++5!#>" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "8th description of " }{XPPEDIT 18 0 "de lta(x)" "6#-%&deltaG6#%\"xG" }{TEXT -1 24 " as a limit of functions" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {XPPEDIT 18 0 "delta(x)" "6#-%&deltaG6#%\"xG" }{TEXT -1 59 " can be re alised as the limit of the sequence of functions " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "s[n](x) = n*sin^2*n*x/(Pi*x^2);" "6#/ -&%\"sG6#%\"nG6#%\"xG*,F(\"\"\"*$%$sinG\"\"#F,F(F,F*F,*&%#PiGF,*$F*F/F ,!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " } {XPPEDIT 18 0 "Int((sin*x/x)^2,x = -infinity .. infinity) = Pi;" "6#/- %$IntG6$*$*(%$sinG\"\"\"%\"xGF*F+!\"\"\"\"#/F+;,$%)infinityGF,F1%#PiG " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Int((sin(x)/x)^2,x=-infinity..infinity);\nval ue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&*$)-%$sinG6#%\"xG \"\"#\"\"\"F.*$)F,F-F.!\"\"/F,;,$%)infinityGF1F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%#PiG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Letting \+ " }{XPPEDIT 18 0 "u = n*x" "6#/%\"uG*&%\"nG\"\"\"%\"xGF'" }{TEXT -1 11 " shows that" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "I nt(sin^2*n*x/(n*Pi*x^2),x = -infinity .. infinity) = 1/Pi;" "6#/-%$Int G6$**%$sinG\"\"#%\"nG\"\"\"%\"xGF+*(F*F+%#PiGF+F,F)!\"\"/F,;,$%)infini tyGF/F3*&F+F+F.F/" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int((sin*u/u)^2,u = -infinity .. infinity) = 1;" "6#/-%$IntG6$*$*(%$sinG\"\"\"%\"uGF*F+! \"\"\"\"#/F+;,$%)infinityGF,F1F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 25 "for any positive integer " }{TEXT 273 1 "n" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "Int(sin(n*x)^2/(n*Pi*x^2),x=-infinity..infinity);\nas sume(n_>0):\nsubs(n_=n,value(subs(n=n_,%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&*$)-%$sinG6#*&%\"nG\"\"\"%\"xGF.\"\"#F.F.*(F -F.%#PiGF.)F/F0F.!\"\"/F/;,$%)infinityGF4F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "s := (n,x) -> sin(n*x)^2/(n*Pi*x^2 ):\nplot([s(1,x),s(2,x),s(3,x),s(4,x)],x=-4..4,\n color =[red,green,blue,magenta]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 270 270 {PLOTDATA 2 "6(-%'CURVESG6$7en7$$!\"%\"\"!$\"1)eGV&*\\%R6!#<7$$!1nmmmF iDQ!#:$\"1?TD(y[]o)!#=7$$!1LLLo!)*Qn$F1$\"1S(y4$4TugF47$$!1nmmwxE.NF1$ \"1X*3J.*GZKF47$$!1mmmOk]JLF1$\"1#H$fJy+A5F47$$!1MLL[9cgJF1$\"1dgBURVY 6!#?7$$!1nmmhN2-IF1$\"1b_I)>R0$o!#>7$$!1+++N&oz$GF1$\"1o/*Qp8G`$F47$$! 1nmm\")3DoEF1$\"1uI'4wr5H*F47$$!1+++:v2*\\#F1$\"1K[\"\\&f**H=F-7$$!1LL L8>1DBF1$\"1')Q3GeGFJF-7$$!1nmmw))yr@F1$\"1Q\"y#y7x!f%F-7$$!1+++S(R#** >F1$\"1=L:%4B#*e'F-7$$!1++++@)f#=F1$\"1p[!Gax%Q*)F-7$$!1+++gi,f;F1$\"1 &p,n.Jv9\"!#;7$$!1nmm\"G&R2:F1$\"1$\\W[,S_R\"Fbp7$$!1LLLtK5F8F1$\"1r3m 5*>@q\"Fbp7$$!1MLL$HsV<\"F1$\"1$RH9&=\"R'>Fbp7$$!1-++]&)4n**Fbp$\"1!fQ 5.s\"fAFbp7$$!1PLLL\\[%R)Fbp$\"1v[wU)QA]#Fbp7$$!1)*****\\&y!pmFbp$\"1d ,l%> ^JFbp7$$!1nm;aH-88Fbp$\"16kVS$[[;$Fbp7$$!11++]-6&)))F-$\"1URjH8tuJFbp7 $$!1SLLe4**RYF-$\"1h!y[;:3=$Fbp7$$!1Tnmmmr[RF4$\"1(*4'yJ#3$=$Fbp7$$\"1 6LL3Uh9SF-$\"1?2(*[\"*Q\"=$Fbp7$$\"1(HLL3+TU)F-$\"1'3ZMJwb<$Fbp7$$\"1G L$efeLG\"Fbp$\"1w-X\")=mlJFbp7$$\"1FLL$=2Vs\"Fbp$\"1`G)Glw;:$Fbp7$$\"1 hmmm7+#\\#Fbp$\"1#R5(Q6v2$Fbp7$$ \"1HLLLm&z\"\\Fbp$\"1%)Guh&3Y$HFbp7$$\"1(******z-6j'Fbp$\"1CY(QIdIu#Fb p7$$\"1#******4#32$)Fbp$\"1Vn$Qi`]^#Fbp7$$\"1%*****\\#y'G**Fbp$\"1LhFbp7$$\"1mmm1>qM8F1$\"1a'R< Wh!*o\"Fbp7$$\"1++++.W2:F1$\"1&)=&Hlk^R\"Fbp7$$\"1LLLep'Rm\"F1$\"1'[A( 3hoR6Fbp7$$\"1+++S>4N=F1$\"1e;E?h?2))F-7$$\"1mmm6s5'*>F1$\"1r#*z#G$yGm F-7$$\"1+++lXTk@F1$\"10;mL'3(oYF-7$$\"1mmmmd'*GBF1$\"1pXaf+%R4$F-7$$\" 1+++DcB,DF1$\"1_@&G66j\"=F-7$$\"1MLLt>:nEF1$\"1;HcRImQ$*F47$$\"1LLL.a# o$GF1$\"1`JB4R]hNF47$$\"1nmm^Q40IF1$\"1/R%oC(oElFO7$$\"1+++!3:(fJF1$\" 1!=5+^#)p/\"FI7$$\"1nmmc%GpL$F1$\"1;s,EY$p2\"F47$$\"1LLL8-V&\\$F1$\"1C MGJ4#z7$F47$$\"1+++XhUkOF1$\"1V6ZA@_5fF47$$\"1+++:o!*F47$F;$\"1#)fbbAV\"o&F47$ F@$\"1.L\"[rv6(>F47$FE$\"1;2wuH/#H#FI7$FK$\"1_0[eyoR8F47$FQ$\"121Rwb.M kF47$FV$\"1^FLWl0s9F-7$Fen$\"1p90@\"eeM#F-7$$!1nm;9(p?T#F1$\"1\\b4@O], FF-7$Fjn$\"1)=B9p!pKHF-7$$!1+++&RD%[AF1$\"1$z;Uu,U+$F-7$F_o$\"1,!R-ymd $HF-7$$!1LLL3V^&3#F1$\"1;$)[+if(o#F-7$Fdo$\"1,ZN&*>kuAF-7$Fio$\"1`)zX; #4R6F-7$$!1+++!=*\\Uc\"y\"F47$$!1nmT:56 @;F1$\"1s%3BY@>6'FO7$$!1LL$3x0Ke\"F1$\"1!zSm<(R5RFI7$$!1++DE0IX:F1$\"1 P!H:Wq9t\"FO7$Fdp$\"1r7jI]=?6F47$$!1LLezAKi9F1$\"1cGK$HH%[MF47$$!1++]x #\\sT\"F1$\"1'4`o741C(F47$$!1nmTvi )>F-7$$!1MLL$yP2D\"F1$\"1UQ+\"R'eHOF-7$F^q$\"1HWa9b+ceF-7$$!1nm;u5a&3 \"F1$\"1IV[t=U'>*F-7$Fcq$\"1%y$e< Fbp7$Fhq$\"140L&orAB#Fbp7$$!1ommTA?4 %o6Q$Fbp7$$!1*******4\"eZeFbp$\"1!)f$)[?LWRFbp7$Fbr$\"1W>P/N)3\\%Fbp7$ $!1mmmTs$*oTFbp$\"1-$RFk'4@]Fbp7$Fgr$\"1k'Qho$*z[&Fbp7$F\\s$\"1V&y4HBK %eFbp7$Fas$\"1^U,Db-9hFbp7$Ffs$\"19#pu`'>@iFbp7$F[t$\"1EPNx%o%*H'Fbp7$ $!1tm;/1binF-$\"1>RL$3Fr)4=F-$\"1wc'[wiF-$\"1&=#pNCVLjFbp7$F_u$\"1K*>Y*y=1j Fbp7$Fdu$\"1\\)pN],v W)[gXFbp7$$\"1jmm;(HXx&Fbp$\"1$QR,g5R*RFbp7$F]w$\"1xusS$Fbp7$$\"1% *****\\C4puFbp$\"1PC4DY,OGFbp7$Fbw$\"1)\\a3/^uG#Fbp7$$\"1$****\\$3$*F-7$F\\x$\"1_MV\"4lT!fF-7$$\"1LLLo,\"QD\"F1$\"14^(yUGBb$F-7$Fax$ \"1d4l*)yD[=F-7$$\"1+++0l)yP\"F1$\"1ImsSw9(=\"F-7$$\"1LLL.62@9F1$\"1Z4 P@%G$eoF47$$\"1mmm,dDk9F1$\"1a^o/Dm>LF47$Ffx$\"1\\Dj9l`=6F47$$\"1LLek> dY:F1$\"1')=mrrkg:FO7$$\"1mm;HOq&e\"F1$\"19.L#pzZi&FI7$$\"1++v$HN[i\"F 1$\"1emT(e#H9qFO7$F[y$\"1(*f%H)4&H(>F47$$\"1mm;\\%H&\\\\$ **pom#F-7$Fjy$\"12fjBdc@HF-7$$\"1LL$e;!pYAF1$\"1pt&=IPU+$F-7$F_z$\"1?X k[_`DHF-7$$\"1LL$ep+^T#F1$\"16-[6J'4p#F-7$Fdz$\"1L@,.D$eL#F-7$Fiz$\"17 +:V*HzZ\"F-7$F^[l$\"1o2F47$F/$\"1Yvq];9'p&F47$F6$\"1J:e^9ebyF47$F;$\"1P[kjVxgnF47$ F@$\"1PFfwbe\"y#F47$FE$\"13XFzH+OMFI7$FK$\"1Ex=tV%[%>F47$FQ$\"1()*G+QF EA)F47$FV$\"1@6*Hx%oc9F-7$$!1LLL)>kOe#F1$\"13%4sO'os:F-7$Fen$\"1*>&=>^ q\"\\\"F-7$F\\`l$\"1Crr#yU=@\"F-7$Fjn$\"1(z9S=L6*zF47$Fd`l$\"1vl&yh+:< %F47$F_o$\"1`+>C/0\">\"F47$F\\al$\"1+\"f/a%>JF1$\"10=D/KU/yF47$Fio$\"1c\\6WGDa;F-7$F^p$\"11!y'R(\\7f$F-7 $F_bl$\"1D%4fHAh%RF-7$Fdbl$\"1W7FzJ>FUF-7$Fibl$\"1ZDjm(GtT%F-7$Fdp$\"1 2p)Q=TE]%F-7$Facl$\"1/mC,6naWF-7$Ffcl$\"1O&=E6Y'QUF-7$F[dl$\"1i$Rs$\\G gQF-7$Fip$\"15_.)QR)QLF-7$Fcdl$\"1k%)z\"G\\)HAF-7$F^q$\"1FaF[Bcm5F-7$$ !1++v$oc*H6F1$\"19Htg83=]F47$F[el$\"1;(p<$)oh=\"F47$$!1+]PpKLj5F1$\"1> (H7s,s>#FO7$$!1MLeka7T5F1$\"1b`A[(GyC$FI7$$!1n;zfw\"*=5F1$\"1ix#zb&[Qt FO7$Fcq$\"1MS\"ozR:V#F47$Fcel$\"1#zI'ps?'z\"F-7$Fhq$\"10*>Rn\"fH^F-7$F [fl$\"1_\\tKg'[6\"Fbp7$F]r$\"1k;O\\q:r>Fbp7$$!1*****\\#)H$eiFbp$\"1d/q m83iCFbp7$Fcfl$\"1\"*z]B'p'**HFbp7$$!1*****\\PKoV&Fbp$\"1ZRLF\"4ld$Fbp 7$Fbr$\"1%*\\\\&fhN=%Fbp7$$!1LL$eW5vf%Fbp$\"1]\\#3hPy$[Fbp7$F[gl$\"1I, kvAQ+bFbp7$$!1****\\PSOSPFbp$\"1+r&)>h(p:'Fbp7$Fgr$\"1j:M[(QGz'Fbp7$$! 1LL$e%oA=HFbp$\"1U,A!z\\dM(Fbp7$F\\s$\"1GTKOyQdyFbp7$$!1LL$3())4J@Fbp$ \"1f_]))G5<$)Fbp7$Fas$\"1AQ_A:9:()Fbp7$F[t$\"1+T_upED$*Fbp7$Fet$\"1'Hm b!*\\)[&*Fbp7$F_u$\"1JPU=*=xM*Fbp7$Fiu$\"15IG\"Gy#*>F-7$Fgw$\"1*z?_'[4MGF47$$ \"1**\\i]TP:5F1$\"1$o(eq6u^$*FO7$$\"1***\\iZ!)y.\"F1$\"1.kf!)*yMp(FI7$ $\"1**\\(=!oQg5F1$\"1gs#)Hydw9FO7$F[]m$\"1c_gxujL5F47$$\"1***\\(yd!z7 \"F1$\"1cx-g;m%z%F47$F\\x$\"1uIy*zlg/\"F-7$Fc]m$\"1\\X(ykvvF#F-7$Fax$ \"1Dn@I@_NMF-7$F[^m$\"16\\ni6n;RF-7$F`^m$\"1sO;L.VjUF-7$$\"1nmm,dDk9F1 $\"1*zmF1$\"1(R_.HM7b(F47$Fey$\"1hO#eb`*[AF47$Fj`m$\"1HF@v$RdS%F I7$Fjy$\"1%=%fs?hZ)*FO7$Fbam$\"1Yc\"ehb44%F47$F_z$\"1)*Q*yBa**=)F47$Fj am$\"1F$evvsVA\"F-7$Fdz$\"1*)H,iXE'\\\"F-7$$\"1nm;*z$>%e#F1$\"1@!)=*=a Dd\"F-7$Fiz$\"1Q$\\BDI$f9F-7$F^[l$\"1*G%p\"G:IF)F47$Fc[l$\"1O,(fa=D'=F 47$Fh[l$\"13vv8w>QJFI7$F]\\l$\"18WXJFQ9HF47$Fb\\l$\"1=0(f5s)>mF47$Fg\\ l$\"1')4seEf,zF47$F\\]l$\"1*zPrH%f%o&F47$Fa]lF`dm-Fd]l6&Ff]lF*F*Fg]l-F $6$7gr7$F($\"1^iL;o_ATFO7$F/$\"1KX`H]5g%)FO7$F6$\"1DWW]=`SUF47$F;$\"1e zaVhm%Q'F47$F@$\"14Om'4B/S$F47$FE$\"1-qW>9\\xXFI7$FK$\"1QgGPF47$F\\`l$\"1Tj?UEK> nFO7$Fjn$\"1Hy(G[T5F#FO7$Fd`l$\"1E$pE0&>F-7$$!1+++I`#f&>F1$\"1N]#z%**>y?F- 7$Fegm$\"1o$3@.0l3#F-7$$!1+++5lHp=F1$\"1qh'e2c!p>F-7$Fio$\"1,GY'yKXt\" F-7$Fgal$\"1ELkn;s`5F-7$F^p$\"1FSfWkM`MF47$F_bl$\"14VWAy/57F47$Fdbl$\" 1*Q$H6z(f\"yFI7$Fibl$\"1:bbTX%RX$FO7$Fdp$\"1vz'zJSX?#F47$Ffcl$\"1(H`$y Nz:8F-7$Fip$\"1!e>t(z_%4$F-7$Fcdl$\"1%RUerH%pYF-7$F^q$\"1Zgo!Rb(odF-7$ $!1n;a)[k@:\"F1$\"1aC0?9MIfF-7$F^jm$\"16$olw)H/gF-7$$!1M$e*y)[x5\"F1$ \"1>f)Hpz[)fF-7$F[el$\"1!GZRVF*oeF-7$F[[n$\"1`knvse\\`F-7$Fcq$\"1GW``5 X$[%F-7$Fcel$\"1BeF(*4\")=CF-7$Fhq$\"1!H_!pW'o>&F47$$!1pmTN\"4)y\")Fbp $\"1q_\\Eg0(*>F47$$!1-+]PL8jzFbp$\"1Slsy;r!R#FO7$$!1OLeRvXZxFbp$\"1%on `N1cS#FO7$F[fl$\"1-bN)*)yrJ#F47$$!1LL$e9I/5(Fbp$\"1hx<#R]6R\"F-7$F]r$ \"1\">%e0H:FPF-7$Fd\\n$\"1V\">fz%H7sF-7$Fcfl$\"1\\dAXDa.7Fbp7$F\\]n$\" 1[\"pvuQR#=Fbp7$Fbr$\"14fF*[8&zDFbp7$Fd]n$\"1+pVT^ cy.zFbp7$F\\s$\"1NE))=:n^*)Fbp7$F\\_n$\"1h\"*R0c;K**Fbp7$Fas$\"17]*H3( )43\"F17$Ffs$\"1-A/Z%*Qg6F17$F[t$\"15yM)pD0A\"F17$F`t$\"1+#e:n'oe7F17$ Fet$\"1hZemO8t7F17$$\"1BK$3-)*\\2(F4$\"1Uu%Hn**GF\"F17$F_il$\"1U'HHu;5 F\"F17$$\"1:LeRFC7HF-$\"1drjI2\\n7F17$Fjt$\"1.gC#fKBE\"F17$Fgil$\"1Z$) fx*)=Z7F17$F_u$\"1z1RMLxD7F17$Fdu$\"1R$yd9a_;\"F17$Fiu$\"12#p(*)*=P3\" F17$F``n$\"1&3:)\\&*[')**Fbp7$F^v$\"12?LWR5O!*Fbp7$Fh`n$\"1-!G>fb(=!)F bp7$Fcv$\"1')po*>#>opFbp7$F`an$\"1!o?.%*3X$eFbp7$F[[m$\"13#z>Fbp7$Fc [m$\"1c&)G_(pPI\"Fbp7$Fhbn$\"15@^1CJ%y(F-7$F]w$\"1q`'\\PZc*RF-7$$\"1'* ***\\i(4]qFbp$\"1D9ST+9*f\"F-7$F[\\m$\"1KHzqqPaLF47$$\"1%***\\i)*eywFb p$\"1+fgT*G@j'FO7$$\"1$****\\F(3))yFbp$\"1k&*=4N3!Q#FI7$$\"1#***\\(o%e (4)Fbp$\"1fB&*e]l[6F47$Fbw$\"1Y@KXRfF-7$F\\x$\"1_8T@v'=y&F-7$Fc]m$\"11\"e#zmy6YF-7$Fax$\"1GSd;QzJHF- 7$F`^m$\"1Iz))*R+BD\"F-7$Ffx$\"1pR$>`Y8?#F47$F]_m$\"1UomDN(R6$FO7$Fb_m $\"1-*3%\\i&R7\"FO7$Fg_m$\"12Mabc`'Q\"F47$F[y$\"1B?%Hnm/\"QF47$F_`m$\" 1grEM*Gt6\"F-7$F`y$\"16&)=V&3Ez\"F-7$$\"1nm\"zvX`(=F1$\"1vO8Mn&G*>F-7$ Fign$\"1[BG$Gy**3#F-7$$\"1++v$R`e&>F1$\"1:bjAkJy?F-7$Fey$\"1];$p(3Tj>F -7$Fj`m$\"1WL>;A2m9F-7$Fjy$\"1!R>XgKH=)F47$Fbam$\"1(z%\\#Qoh$GF47$F_z$ \"1d)=Xv'[LNgWF47 $F\\]l$\"1P@([B30P)FO7$Fa]lFf[o-Fd]l6&Ff]lFg]lF*Fg]l-%+AXESLABELSG6$Q \"x6\"%!G-%%VIEWG6$;F(Fa]l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Int(s(10000,x),x=-1/100..1/100);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*&*$)-%$sinG6#,$%\"xG\"&++\"\"\"#\"\"\"F1*&% #PiGF1)F.F0F1!\"\"#F1F//F.;#F5\"$+\"#F1F:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+b(3$o**!#5" }}}{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 49 "The sam pling property of the Dirac delta function" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 25 "The Dirac delta f unction " }{XPPEDIT 18 0 "delta(x)" "6#-%&deltaG6#%\"xG" }{TEXT -1 26 " is available in Maple as " }{TEXT 0 5 "Dirac" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "delta(x)" "6#-%&deltaG6 #%\"xG" }{TEXT -1 41 " has the property that, for any function " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 25 ", which is contin uous at " }{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 16 ", and s uch that " }{XPPEDIT 18 0 "f(x) -> 0" "6#f*6#-%\"fG6#%\"xG7\"6$%)opera torG%&arrowG6\"\"\"!F-F-F-" }{TEXT -1 4 " as " }{XPPEDIT 18 0 "x -> in finity" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\"%)infinityGF*F*F*" } {TEXT -1 4 " or " }{XPPEDIT 18 0 "x -> -infinity" "6#f*6#%\"xG7\"6$%)o peratorG%&arrowG6\",$%)infinityG!\"\"F*F*F*" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(delta(x)*f(x),x = -infi nity .. infinity)=f(0)" "6#/-%$IntG6$*&-%&deltaG6#%\"xG\"\"\"-%\"fG6#F +F,/F+;,$%)infinityG!\"\"F3-F.6#\"\"!" }{TEXT -1 14 " ------- (i)." } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 262 12 "____________" }{TEXT -1 18 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 54 "Since we obtain a single sample value of the funct ion " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 57 " by means \+ of evaluating the integral, this is called the " }{TEXT 261 17 "sampli ng property" }{TEXT -1 29 " of the Dirac delta function." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "f := \+ 'f':\nalias(delta=Dirac):\nInt(delta(x)*f(x),x=-infinity..infinity);\n value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%&deltaG6#%\" xG\"\"\"-%\"fGF)F+/F*;,$%)infinityG!\"\"F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"fG6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 4 "Note" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 42 "We may interpr et equation (i) to mean that" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Limit(``,n=infinity)" "6#-%&LimitG6$%!G/%\"nG%)infinity G" }{XPPEDIT 18 0 "Int(d[n](x)*f(x),x=-infinity..infinity) = f(0)" "6# /-%$IntG6$*&-&%\"dG6#%\"nG6#%\"xG\"\"\"-%\"fG6#F.F//F.;,$%)infinityG! \"\"F6-F16#\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "d[n](x)" "6#-&%\"dG6#%\"nG6#%\"xG" }{TEXT -1 45 " i s a sequence of functions which represents " }{XPPEDIT 18 0 "delta(x) " "6#-%&deltaG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{TEXT 274 1 "n" }{TEXT -1 29 " \+ be a large positive integer." }}{PARA 0 "" 0 "" {TEXT -1 21 "Then we c an think of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 49 " a s being approximately constant on the interval " }{XPPEDIT 18 0 "[-1/n ,1/n]" "6#7$,$*&\"\"\"F&%\"nG!\"\"F(*&F&F&F'F(" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 9 "Consider " }{XPPEDIT 18 0 "delta(x)" "6#-% &deltaG6#%\"xG" }{TEXT -1 45 " to be the limit of the sequence of func tions" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d[n](x) = PI ECEWISE([n/2, abs(x) < 1/n],[0, 1/n <= abs(x)])" "6#/-&%\"dG6#%\"nG6#% \"xG-%*PIECEWISEG6$7$*&F(\"\"\"\"\"#!\"\"2-%$absG6#F**&F0F0F(F27$\"\"! 1*&F0F0F(F2-F56#F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "Then " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(delta(x)*f(x ),x = -infinity .. infinity)" "6#-%$IntG6$*&-%&deltaG6#%\"xG\"\"\"-%\" fG6#F*F+/F*;,$%)infinityG!\"\"F2" }{TEXT -1 3 " " }{TEXT 263 1 "~" } {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(d[n](x)*f(x),x=-1/n..1/n)" "6#-%$I ntG6$*&-&%\"dG6#%\"nG6#%\"xG\"\"\"-%\"fG6#F-F./F-;,$*&F.F.F+!\"\"F6*&F .F.F+F6" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = n/2;" "6#/%!G*&%\"nG\"\"\"\"\"#!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int(f(x),x=-1/n..1/n)" "6#-%$IntG6$-%\"fG6#%\"xG/F);,$* &\"\"\"F.%\"nG!\"\"F0*&F.F.F/F0" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{TEXT 264 1 "~" }{TEXT -1 2 " " }{XPPEDIT 18 0 "n/2 " "6#*&%\"nG\"\"\"\"\"#!\"\"" }{XPPEDIT 18 0 "``(2*f(0)/n) = f(0);" "6 #/-%!G6#*(\"\"#\"\"\"-%\"fG6#\"\"!F)%\"nG!\"\"-F+6#F-" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Here \+ is a numerical example to illustrate this." }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x) = ln((x+5)^2)*cos(x+5)/(x^2+1);" "6#/ -%\"fG6#%\"xG*(-%#lnG6#*$,&F'\"\"\"\"\"&F.\"\"#F.-%$cosG6#,&F'F.F/F.F. ,&*$F'F0F.F.F.!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "f := x -> ln((x+5)^2)*cos( x+5)/(x^2+1);\n'f'(0)=evalf(f(0));\nn := 10^6;\nn/2*Int('f'(x),x=-1/n. .1/n);\nevalf(%);\nn := 'n':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG f*6#%\"xG6\"6$%)operatorG%&arrowGF(*(-%#lnG6#*$),&9$\"\"\"\"\"&F4\"\"# F4F4-%$cosG6#F2F4,&*$)F3F6F4F4F4F4!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#\"\"!$\"+9NtI\"*!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"(+++\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*& \"'++]\"\"\"-%$IntG6$-%\"fG6#%\"xG/F-;#!\"\"\"(+++\"#F&F2F&F&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+5NtI\"*!#5" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 104 "To indicate that the the result does not depend o n the limiting sequence of functions we use to realise " }{XPPEDIT 18 0 "delta(x)" "6#-%&deltaG6#%\"xG" }{TEXT -1 72 ", we consider another \+ numerical example, using the sequence of functions" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "b[n](x) = n/sqrt(2*Pi);" "6#/-&%\"bG6 #%\"nG6#%\"xG*&F(\"\"\"-%%sqrtG6#*&\"\"#F,%#PiGF,!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "exp(-n^2*x^2/2);" "6#-%$expG6#,$*(%\"nG\"\"#%\"xGF)F )!\"\"F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " } {XPPEDIT 18 0 "f(x) = 2/(1+exp(x^2)*arctan(x+2));" "6#/-%\"fG6#%\"xG*& \"\"#\"\"\",&F*F**&-%$expG6#*$F'F)F*-%'arctanG6#,&F'F*F)F*F*F*!\"\"" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "b := (n,x) -> n*exp(-n^2*x^2/2)/sqrt(2*Pi);\nf \+ := x -> 2/(1+exp(x^2)*arctan(x+2));\n'f'(0)=evalf(f(0));\nn := 3*10^3; \nInt('b'(n,x)*'f'(x),x=-infinity..infinity);\nevalf(%);\nn := 'n':" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bGf*6$%\"nG%\"xG6\"6$%)operatorG% &arrowGF)*(9$\"\"\"-%$expG6#,$*&#F/\"\"#F/*&)F.F6F/)9%F6F/F/!\"\"F/-%% sqrtG6#,$*&F6F/%#PiGF/F/F;F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,$*&\"\"#\"\"\",&F/F/*&-%$expG6 #*$)9$F.F/F/-%'arctanG6#,&F7F/F.F/F/F/!\"\"F/F(F(F(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"fG6#\"\"!$\"+C!*\\\"\\*!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"%+I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6 $*&-%\"bG6$\"%+I%\"xG\"\"\"-%\"fG6#F+F,/F+;,$%)infinityG!\"\"F3" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+?&)\\\"\\*!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 69 "The Dirac delta function as the limit o f a general class of functions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 98 "There are many possible sequen ces of functions which can be used to model the Dirac delta function" }}{PARA 0 "" 0 "" {TEXT -1 39 "In fact we can start with any function \+ " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 11 " such that " } }{PARA 0 "" 0 "" {TEXT -1 8 " (i) " }{XPPEDIT 18 0 "0 <= g(x);" "6# 1\"\"!-%\"gG6#%\"xG" }{TEXT -1 9 " for all " }{TEXT 275 1 "x" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 7 " (ii) " }{XPPEDIT 18 0 "Int(g( x),x = -infinity .. infinity) = 1;" "6#/-%$IntG6$-%\"gG6#%\"xG/F*;,$%) infinityG!\"\"F.\"\"\"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 7 " (iii) " }{XPPEDIT 18 0 "g(-x) = g(x);" "6#/-%\"gG6#,$%\"xG!\"\"-F%6# F(" }{TEXT -1 9 " for all " }{TEXT 276 1 "x" }{TEXT -1 11 ", that is, \+ " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 21 " is an even fu nction." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "The last condition is not essential, but it simplifies the discuss ion." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "C onsider the sequence of functions " }{XPPEDIT 18 0 "g[n](x);" "6#-&%\" gG6#%\"nG6#%\"xG" }{TEXT -1 10 " given by " }{XPPEDIT 18 0 "g[n](x) = \+ n*g(n*x);" "6#/-&%\"gG6#%\"nG6#%\"xG*&F(\"\"\"-F&6#*&F(F,F*F,F," } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 8 "Letting " }{XPPEDIT 18 0 "u=n*x" "6#/%\"uG*&%\"nG\"\"\"%\"xGF'" }{TEXT -1 5 " in " }{XPPEDIT 18 0 "Int(g[n](x),x = -infinity .. infinity) = Int(n*g(n*x),x = -infin ity .. infinity);" "6#/-%$IntG6$-&%\"gG6#%\"nG6#%\"xG/F-;,$%)infinityG !\"\"F1-F%6$*&F+\"\"\"-F)6#*&F+F6F-F6F6/F-;,$F1F2F1" }{TEXT -1 13 ", s hows that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(g[n ](x),x = -infinity .. infinity) = Int(g(u),u = -infinity .. infinity); " "6#/-%$IntG6$-&%\"gG6#%\"nG6#%\"xG/F-;,$%)infinityG!\"\"F1-F%6$-F)6# %\"uG/F7;,$F1F2F1" }{XPPEDIT 18 0 " ``= 1" "6#/%!G\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Sinc e " }{XPPEDIT 18 0 "Int(g(x),x = -infinity .. infinity) = 1;" "6#/-%$I ntG6$-%\"gG6#%\"xG/F*;,$%)infinityG!\"\"F.\"\"\"" }{TEXT -1 23 ", we c an always choose " }{TEXT 277 1 "R" }{TEXT -1 22 " large enough so tha t " }{XPPEDIT 18 0 "Int(g(x),x = R .. infinity) < epsilon/2;" "6#2-%$I ntG6$-%\"gG6#%\"xG/F*;%\"RG%)infinityG*&%(epsilonG\"\"\"\"\"#!\"\"" } {TEXT -1 42 ", for any specified small positive number " }{XPPEDIT 18 0 "epsilon" "6#%(epsilonG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 7 "Then " }{XPPEDIT 18 0 "1-epsilon < Int(g(x),x = -R .. R);" "6#2,& \"\"\"F%%(epsilonG!\"\"-%$IntG6$-%\"gG6#%\"xG/F.;,$%\"RGF'F2" } {XPPEDIT 18 0 "`` <= 1;" "6#1%!G\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Letting " }{XPPEDIT 18 0 "u=n*x" "6#/%\"uG*&%\"nG\"\"\"%\"xGF'" }{TEXT -1 5 " in " } {XPPEDIT 18 0 "Int(g[n](x),x = R/n .. infinity) = Int(n*g(n*x),x = R/n .. infinity);" "6#/-%$IntG6$-&%\"gG6#%\"nG6#%\"xG/F-;*&%\"RG\"\"\"F+! \"\"%)infinityG-F%6$*&F+F2-F)6#*&F+F2F-F2F2/F-;*&F1F2F+F3F4" }{TEXT -1 13 ", shows that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(g[n](x),x = R/n .. infinity) = Int(g(u),u = R .. infinity);" "6 #/-%$IntG6$-&%\"gG6#%\"nG6#%\"xG/F-;*&%\"RG\"\"\"F+!\"\"%)infinityG-F% 6$-F)6#%\"uG/F9;F1F4" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "He nce " }{XPPEDIT 18 0 "1-epsilon < Int(g[n](x),x = -R/n .. R/n);" "6#2, &\"\"\"F%%(epsilonG!\"\"-%$IntG6$-&%\"gG6#%\"nG6#%\"xG/F1;,$*&%\"RGF%F /F'F'*&F6F%F/F'" }{XPPEDIT 18 0 "`` <= 1;" "6#1%!G\"\"\"" }{TEXT -1 10 ", for all " }{TEXT 278 1 "n" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 8 "Even if " }{TEXT 279 1 "R" }{TEXT -1 32 " is large, we can always choose " }{TEXT 280 1 "n" }{TEXT -1 22 " large enough so that \+ " }{XPPEDIT 18 0 "R/n < epsilon" "6#2*&%\"RG\"\"\"%\"nG!\"\"%(epsilonG " }{TEXT -1 15 ", in which case" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "1-epsilon < Int(g[n](x),x = -epsilon .. epsilon);" "6#2 ,&\"\"\"F%%(epsilonG!\"\"-%$IntG6$-&%\"gG6#%\"nG6#%\"xG/F1;,$F&F'F&" } {XPPEDIT 18 0 "`` <= 1;" "6#1%!G\"\"\"" }{TEXT -1 1 "." }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{TEXT 265 13 "_____________" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Thus, for any such sequence " }{XPPEDIT 18 0 "g[n](x);" "6#-&%\"gG6#%\"nG6#%\"x G" }{TEXT -1 74 ", the area under the graph becomes progressively more concentrated around " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Examples " } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 72 "The examples considered in the previous section are obtained as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "1. " }{XPPEDIT 18 0 "g(x)=PIECEWISE([1/2 , abs(x)<1],[0 , abs(x)>=1 ])" "6#/-%\"gG6#%\"xG-%*PIECEWISEG6$7$*&\"\"\"F-\"\"#!\"\"2-%$absG6#F' F-7$\"\"!1F--F26#F'" }{TEXT -1 21 " gives the sequence " }{XPPEDIT 18 0 "g[n](x)=PIECEWISE([n/2 , abs(x)<1/n],[0 , abs(x)>=1/n])" "6#/-&% \"gG6#%\"nG6#%\"xG-%*PIECEWISEG6$7$*&F(\"\"\"\"\"#!\"\"2-%$absG6#F**&F 0F0F(F27$\"\"!1*&F0F0F(F2-F56#F*" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "2. " }{XPPEDIT 18 0 "g(x) =PIECEWISE([1-abs(x) , abs(x)<1],[0 , abs(x)>=1])" "6#/-%\"gG6#%\"xG-% *PIECEWISEG6$7$,&\"\"\"F--%$absG6#F'!\"\"2-F/6#F'F-7$\"\"!1F--F/6#F'" }{TEXT -1 21 " gives the sequence " }{XPPEDIT 18 0 "g[n](x)=PIECEWISE ([n-n^2*abs(x) , abs(x)<1/n],[0 , abs(x)>=1/n" "6#/-&%\"gG6#%\"nG6#%\" xG-%*PIECEWISEG6$7$,&F(\"\"\"*&F(\"\"#-%$absG6#F*F0!\"\"2-F46#F**&F0F0 F(F67$\"\"!1*&F0F0F(F6-F46#F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "3. " }{XPPEDIT 18 0 "g(x)= 1/(Pi*(1+x^2))" "6#/-%\"gG6#%\"xG*&\"\"\"F)*&%#PiGF),&F)F)*$F'\"\"#F)F )!\"\"" }{TEXT -1 21 " gives the sequence " }{XPPEDIT 18 0 "g[n](x)=n /(Pi*(1+n^2*x^2))" "6#/-&%\"gG6#%\"nG6#%\"xG*&F(\"\"\"*&%#PiGF,,&F,F,* &F(\"\"#F*F1F,F,!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "4. " }{XPPEDIT 18 0 "g(x) = exp(-abs(x ))/2;" "6#/-%\"gG6#%\"xG*&-%$expG6#,$-%$absG6#F'!\"\"\"\"\"\"\"#F0" } {TEXT -1 21 " gives the sequence " }{XPPEDIT 18 0 "g[n](x) = n*exp(-n *abs(x))/2;" "6#/-&%\"gG6#%\"nG6#%\"xG*(F(\"\"\"-%$expG6#,$*&F(F,-%$ab sG6#F*F,!\"\"F,\"\"#F5" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "5. " }{XPPEDIT 18 0 "g(x)=1/sqrt(2*Pi )" "6#/-%\"gG6#%\"xG*&\"\"\"F)-%%sqrtG6#*&\"\"#F)%#PiGF)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-x^2/2)" "6#-%$expG6#,$*&%\"xG\"\"#F)!\" \"F*" }{TEXT -1 20 " gives the sequence " }{XPPEDIT 18 0 "g[n](x)=n/sq rt(2*Pi)" "6#/-&%\"gG6#%\"nG6#%\"xG*&F(\"\"\"-%%sqrtG6#*&\"\"#F,%#PiGF ,!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-n^2*x^2/2)" "6#-%$expG6#, $*(%\"nG\"\"#%\"xGF)F)!\"\"F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "6. " }{XPPEDIT 18 0 "g(x) \+ = PIECEWISE([cos(Pi*x/2)^2, abs(x) < 1],[0, 1 <= abs(x)]);" "6#/-%\"gG 6#%\"xG-%*PIECEWISEG6$7$*$-%$cosG6#*(%#PiG\"\"\"F'F2\"\"#!\"\"F32-%$ab sG6#F'F27$\"\"!1F2-F76#F'" }{TEXT -1 21 " gives the sequence " } {XPPEDIT 18 0 "g[n](x) = PIECEWISE([n*cos(n*Pi*x/2)^2, abs(x) < 1/n],[ 0, 1/n <= abs(x)]);" "6#/-&%\"gG6#%\"nG6#%\"xG-%*PIECEWISEG6$7$*&F(\" \"\"*$-%$cosG6#**F(F0%#PiGF0F*F0\"\"#!\"\"F7F02-%$absG6#F**&F0F0F(F87$ \"\"!1*&F0F0F(F8-F;6#F*" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "7. " }{XPPEDIT 18 0 "g(x) = PIECEWI SE([[1/A]*exp(1/(x^2-1)), abs(x) < 1],[0, 1 <= abs(x)]);" "6#/-%\"gG6# %\"xG-%*PIECEWISEG6$7$*&7#*&\"\"\"F/%\"AG!\"\"F/-%$expG6#*&F/F/,&*$F' \"\"#F/F/F1F1F/2-%$absG6#F'F/7$\"\"!1F/-F;6#F'" }{TEXT -1 9 " , where \+ " }{XPPEDIT 18 0 "A =Int(exp(1/(x^2-1)),x=-1..1)" "6#/%\"AG-%$IntG6$-% $expG6#*&\"\"\"F,,&*$%\"xG\"\"#F,F,!\"\"F1/F/;,$F,F1F," }{TEXT -1 20 " , gives the sequence" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "g[n](x) = PIECEWISE([[n/A]*exp(1/(n^2*x^2-1)), abs(x) < 1/n],[0, 1/n <= abs(x)]);" "6#/-&%\"gG6#%\"nG6#%\"xG-%*PIECEWISEG6$7$*&7#*&F( \"\"\"%\"AG!\"\"F2-%$expG6#*&F2F2,&*&F(\"\"#F*F;F2F2F4F4F22-%$absG6#F* *&F2F2F(F47$\"\"!1*&F2F2F(F4-F>6#F*" }{TEXT -1 2 " ." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "8. " }{XPPEDIT 18 0 "g(x ) = sin^2*x/(Pi*x^2);" "6#/-%\"gG6#%\"xG*(%$sinG\"\"#F'\"\"\"*&%#PiGF+ *$F'F*F+!\"\"" }{TEXT -1 21 " gives the sequence " }{XPPEDIT 18 0 "g[ n](x) = sin^2*n*x/(n*Pi*x^2);" "6#/-&%\"gG6#%\"nG6#%\"xG**%$sinG\"\"#F (\"\"\"F*F.*(F(F.%#PiGF.F*F-!\"\"" }{TEXT -1 2 ", " }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Now let " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG " }{TEXT -1 8 " be any " }{TEXT 261 27 "bounded continuous function" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 25 "Note that supposing tha t " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 57 " is identica lly 0 outside of some finite closed interval " }{XPPEDIT 18 0 "[a,b]" "6#7$%\"aG%\"bG" }{TEXT -1 30 " is sufficient to ensure that " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 12 " is bounded." }} {PARA 0 "" 0 "" {TEXT -1 16 "We can show that" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(``,n=infinity)" "6#-%&LimitG6$%!G /%\"nG%)infinityG" }{XPPEDIT 18 0 "``(Int(g[n](x)*f(x),x = -infinity . . infinity)) = f(0);" "6#/-%!G6#-%$IntG6$*&-&%\"gG6#%\"nG6#%\"xG\"\"\" -%\"fG6#F1F2/F1;,$%)infinityG!\"\"F9-F46#\"\"!" }{TEXT -1 3 ". " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 266 19 "___________________" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "This means that the sampling property, stated in this man ner, is " }{TEXT 261 37 "independent of the choice of sequence" } {TEXT -1 1 " " }{XPPEDIT 18 0 "g[n](x)" "6#-&%\"gG6#%\"nG6#%\"xG" } {TEXT -1 19 " used to represent " }{XPPEDIT 18 0 "delta(x)" "6#-%&delt aG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 31 "This leads t o the concept of a " }{TEXT 261 12 "distribution" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 16 "The real number " }{XPPEDIT 18 0 "sqrt(2) " "6#-%%sqrtG6#\"\"#" }{TEXT -1 86 " can be represented by many differ ent sequences of rational numbers which converge to " }{XPPEDIT 18 0 " sqrt(2)" "6#-%%sqrtG6#\"\"#" }{TEXT -1 12 " as a limit." }}{PARA 0 "" 0 "" {TEXT -1 54 "Two examples of such sequences of rational numbers a re" }}{PARA 0 "" 0 "" {TEXT -1 6 " (i) " }{XPPEDIT 18 0 "3/2,7/5,17/1 2,41/29,99/70,239/169,577/408, ` . . . `" "6**&\"\"$\"\"\"\"\"#!\"\"*& \"\"(F%\"\"&F'*&\"# " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "The sampling propert y" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "In t(g[n](x),x = -infinity .. infinity) = 1;" "6#/-%$IntG6$-&%\"gG6#%\"nG 6#%\"xG/F-;,$%)infinityG!\"\"F1\"\"\"" }{TEXT -1 1 "," }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "Int(g[n](x)*f(x),x = -infinity . . infinity)-f(0) = Int(g[n](x)*f(x),x = -infinity .. infinity)-f(0)*In t(g[n](x),x = -infinity .. infinity);" "6#/,&-%$IntG6$*&-&%\"gG6#%\"nG 6#%\"xG\"\"\"-%\"fG6#F/F0/F/;,$%)infinityG!\"\"F7F0-F26#\"\"!F8,&-F&6$ *&-&F+6#F-6#F/F0-F26#F/F0/F/;,$F7F8F7F0*&-F26#F;F0-F&6$-&F+6#F-6#F//F/ ;,$F7F8F7F0F8" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = Int(``(g[n](x)*f(x)-g[n](x)*f(0)),x = -infinity .. infinity);" "6#/%!G-%$IntG6$-F$6#,&*&-&%\"gG6#%\"nG6#%\"xG\"\"\"-%\"f G6#F2F3F3*&-&F.6#F06#F2F3-F56#\"\"!F3!\"\"/F2;,$%)infinityGF?FC" } {TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` \+ = Int((f(x)-f(0))*g[n](x),x = -infinity .. infinity);" "6#/%!G-%$IntG6 $*&,&-%\"fG6#%\"xG\"\"\"-F+6#\"\"!!\"\"F.-&%\"gG6#%\"nG6#F-F./F-;,$%)i nfinityGF2F<" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 2 "so" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "abs(Int(g[n](x)*f(x), x = -infinity .. infinity)-f(0)) = abs(Int((f(x)-f(0))*g[n](x),x = -in finity .. infinity));" "6#/-%$absG6#,&-%$IntG6$*&-&%\"gG6#%\"nG6#%\"xG \"\"\"-%\"fG6#F2F3/F2;,$%)infinityG!\"\"F:F3-F56#\"\"!F;-F%6#-F)6$*&,& -F56#F2F3-F56#F>F;F3-&F.6#F06#F2F3/F2;,$F:F;F:" }{TEXT -1 2 " " }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "``<= Int(abs(f(x)-f( 0))*g[n](x),x=-infinity..infinity)" "6#1%!G-%$IntG6$*&-%$absG6#,&-%\"f G6#%\"xG\"\"\"-F.6#\"\"!!\"\"F1-&%\"gG6#%\"nG6#F0F1/F0;,$%)infinityGF5 F?" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 " Letting " }{XPPEDIT 18 0 "u=n*x" "6#/%\"uG*&%\"nG\"\"\"% \"xGF'" }{TEXT -1 4 " in " }{XPPEDIT 18 0 "Int(abs(f(x)-f(0))*g[n](x), x=0..infinity)" "6#-%$IntG6$*&-%$absG6#,&-%\"fG6#%\"xG\"\"\"-F,6#\"\"! !\"\"F/-&%\"gG6#%\"nG6#F.F//F.;F2%)infinityG" }{TEXT -1 7 " gives " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(abs(f(x)-f(0))*g[ n](x),x=0..infinity)=Int(abs(f(u/n)-f(0))*g(u),u=0..infinity)" "6#/-%$ IntG6$*&-%$absG6#,&-%\"fG6#%\"xG\"\"\"-F-6#\"\"!!\"\"F0-&%\"gG6#%\"nG6 #F/F0/F/;F3%)infinityG-F%6$*&-F)6#,&-F-6#*&%\"uGF0F9F4F0-F-6#F3F4F0-F7 6#FGF0/FG;F3F=" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 "= " } {XPPEDIT 18 0 "Int(abs(f(u/n)-f(0))*g(u),u=0..R)+Int(abs(f(u/n)-f(0))* g(u),u=R..infinity)" "6#,&-%$IntG6$*&-%$absG6#,&-%\"fG6#*&%\"uG\"\"\"% \"nG!\"\"F1-F-6#\"\"!F3F1-%\"gG6#F0F1/F0;F6%\"RGF1-F%6$*&-F)6#,&-F-6#* &F0F1F2F3F1-F-6#F6F3F1-F86#F0F1/F0;F<%)infinityGF1" }{TEXT -1 15 " -- ----- (i). " }}{PARA 0 "" 0 "" {TEXT -1 45 "Consider the 2nd of the tw o integrals in (i)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 27 "Since we have assumed that " }{XPPEDIT 18 0 "f(x)" "6#- %\"fG6#%\"xG" }{TEXT -1 40 " is bounded, there is a positive number " }{TEXT 281 1 "M" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "abs(f(x)) \+ <= M;" "6#1-%$absG6#-%\"fG6#%\"xG%\"MG" }{TEXT -1 9 " for all " } {TEXT 282 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 6 "Then " }{XPPEDIT 18 0 "abs(f(u/n)-f(0)) < = abs(f(u/n))+abs(f(0));" "6#1-%$absG6#,&-%\"fG6#*&%\"uG\"\"\"%\"nG!\" \"F--F)6#\"\"!F/,&-F%6#-F)6#*&F,F-F.F/F--F%6#-F)6#F2F-" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "``<= 2*M" "6#1%!G*&\"\"#\"\"\"%\"MGF'" }{TEXT -1 10 ", for all " }{TEXT 283 1 "u" }{TEXT -1 4 ", so" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(abs(f(u/n)-f(0))*g(u),u=R..infinity ) <= 2*M*Int(g(u),u=R..infinity)" "6#1-%$IntG6$*&-%$absG6#,&-%\"fG6#*& %\"uG\"\"\"%\"nG!\"\"F1-F-6#\"\"!F3F1-%\"gG6#F0F1/F0;%\"RG%)infinityG* (\"\"#F1%\"MGF1-F%6$-F86#F0/F0;F f(0)" "6#f*6#-%\"fG6#%\"xG7\"6$%)operatorG%&arrowG6\"-F&6#\"\"!F-F-F- " }{TEXT -1 4 " as " }{XPPEDIT 18 0 "x->0" "6#f*6#%\"xG7\"6$%)operator G%&arrowG6\"\"\"!F*F*F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 19 "Thus we can choose " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT -1 10 " so that " }{XPPEDIT 18 0 "abs(f(x)-f(0)) < epsilon/2;" "6#2-% $absG6#,&-%\"fG6#%\"xG\"\"\"-F)6#\"\"!!\"\"*&%(epsilonGF,\"\"#F0" } {TEXT -1 10 ", for all " }{TEXT 285 1 "x" }{TEXT -1 6 " with " } {XPPEDIT 18 0 "0 <= x;" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "`` < delta;" " 6#2%!G%&deltaG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 19 "Then we can choose " }{TEXT 286 1 "n" }{TEXT -1 29 " sufficiently large, so t hat " }{XPPEDIT 18 0 "R/n <= delta;" "6#1*&%\"RG\"\"\"%\"nG!\"\"%&delt aG" }{TEXT -1 13 ", which gives" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "abs(f(x)-f(0)) < epsilon/2;" "6#2-%$absG6#,&-%\"fG6#%\" xG\"\"\"-F)6#\"\"!!\"\"*&%(epsilonGF,\"\"#F0" }{TEXT -1 10 ", for all \+ " }{TEXT 287 1 "x" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "0 <= x;" "6#1 \"\"!%\"xG" }{XPPEDIT 18 0 "`` <= R/n;" "6#1%!G*&%\"RG\"\"\"%\"nG!\"\" " }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "abs(f(u/n)-f(0)) < epsilon/ 2;" "6#2-%$absG6#,&-%\"fG6#*&%\"uG\"\"\"%\"nG!\"\"F--F)6#\"\"!F/*&%(ep silonGF-\"\"#F/" }{TEXT -1 10 ", for all " }{TEXT 288 1 "u" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "0 <= u;" "6#1\"\"!%\"uG" }{XPPEDIT 18 0 "`` <= R;" "6#1%!G%\"RG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 15 " It follows that" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "In t(abs(f(u/n)-f(0))*g(u),u = 0 .. R) < epsilon/2;" "6#2-%$IntG6$*&-%$ab sG6#,&-%\"fG6#*&%\"uG\"\"\"%\"nG!\"\"F1-F-6#\"\"!F3F1-%\"gG6#F0F1/F0;F 6%\"RG*&%(epsilonGF1\"\"#F3" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(g(u), u = 0 .. R)<=epsilon/2" "6#1-%$IntG6$-%\"gG6#%\"uG/F*;\"\"!%\"RG*&%(ep silonG\"\"\"\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 24 "We have now shown that " }{XPPEDIT 18 0 "Int(abs(f(x)-f(0))*g[n](x), x=0..infinity)->0" "6#f*6#-%$IntG6$*&-%$absG6#,&-%\"fG6#%\"xG\"\"\"-F. 6#\"\"!!\"\"F1-&%\"gG6#%\"nG6#F0F1/F0;F4%)infinityG7\"6$%)operatorG%&a rrowG6\"F4FCFCFC" }{TEXT -1 4 " as " }{XPPEDIT 18 0 "n->infinity" "6#f *6#%\"nG7\"6$%)operatorG%&arrowG6\"%)infinityGF*F*F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "Similar ly, " }{XPPEDIT 18 0 "Int(abs(f(x)-f(0))*g[n](x),x=-infinity..0)->0" "6#f*6#-%$IntG6$*&-%$absG6#,&-%\"fG6#%\"xG\"\"\"-F.6#\"\"!!\"\"F1-&%\" gG6#%\"nG6#F0F1/F0;,$%)infinityGF5F47\"6$%)operatorG%&arrowG6\"F4FDFDF D" }{TEXT -1 4 " as " }{XPPEDIT 18 0 "n->infinity" "6#f*6#%\"nG7\"6$%) operatorG%&arrowG6\"%)infinityGF*F*F*" }{TEXT -1 5 ", so " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(abs(f(x)-f(0))*g[n](x), x=-infinity..infinity)->0" "6#f*6#-%$IntG6$*&-%$absG6#,&-%\"fG6#%\"xG \"\"\"-F.6#\"\"!!\"\"F1-&%\"gG6#%\"nG6#F0F1/F0;,$%)infinityGF5F?7\"6$% )operatorG%&arrowG6\"F4FDFDFD" }{TEXT -1 4 " as " }{XPPEDIT 18 0 "n->i nfinity" "6#f*6#%\"nG7\"6$%)operatorG%&arrowG6\"%)infinityGF*F*F*" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "This completes the argument to show that" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(``,n=infinity)" "6#-%&LimitG 6$%!G/%\"nG%)infinityG" }{XPPEDIT 18 0 "Int(g[n](x)*f(x),x=-infinity.. infinity) = f(0)" "6#/-%$IntG6$*&-&%\"gG6#%\"nG6#%\"xG\"\"\"-%\"fG6#F. F//F.;,$%)infinityG!\"\"F6-F16#\"\"!" }{TEXT -1 2 ". " }}{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 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }