{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 "Red Emphasis" -1 256 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{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 "" -1 259 "Geneva" 1 9 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 1 10 0 0 0 0 0 0 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 0 }{CSTYLE "Dark Red Emphasis" -1 262 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Blue Emphasis" -1 263 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " Grey Emphasis" -1 266 "Times" 1 12 96 52 84 1 0 1 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 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Time s" 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 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 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 48 "Root-finding for functions of a c omplex variable" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaim o, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 24.3.2007" } }{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 40 "load root-finding procedures incl uding: " }{TEXT 0 10 "interproot" }}{PARA 0 "" 0 "" {TEXT -1 17 "The M aple m-file " }{TEXT 266 7 "roots.m" }{TEXT -1 37 " contains the code \+ for the procedure " }{TEXT 0 10 "interproot" }{TEXT -1 25 " used in th is worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 122 "It can be read into a \+ Maple session by a command similar to the one that follows, where the \+ file path gives its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "read \"K:\\\\Maple/procdrs/roots.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 17 "The zeta function" }}{PARA 0 " " 0 "" {TEXT -1 45 "It can be shown by using Fourier series that " } {XPPEDIT 18 0 "Sum(1/(n^2),n = 1 .. infinity) =1+1/4+1/9+1/16+1/25+` . . . `" "6#/-%$SumG6$*&\"\"\"F(*$%\"nG\"\"#!\"\"/F*;F(%)infinityG,.F(F (*&F(F(\"\"%F,F(*&F(F(\"\"*F,F(*&F(F(\"#;F,F(*&F(F(\"#DF,F(%(~.~.~.~GF (" }{XPPEDIT 18 0 "`` = Pi^2/6;" "6#/%!G*&%#PiG\"\"#\"\"'!\"\"" } {TEXT -1 2 ".\n" }}{PARA 0 "" 0 "" {TEXT -1 24 "Maple knows this resul t." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "sum(1/(n^2),n = 1 .. infinity);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$)%#PiG\"\"#\"\"\"#F(\"\"'" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+oS$\\k\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "For another similar example we have \+ " }{XPPEDIT 18 0 "Sum(1/(n^4),n = 1 .. infinity)=1+1/16+1/81+1/256+1/6 25+` . . . ` " "6#/-%$SumG6$*&\"\"\"F(*$%\"nG\"\"%!\"\"/F*;F(%)infinit yG,.F(F(*&F(F(\"#;F,F(*&F(F(\"#\")F,F(*&F(F(\"$c#F,F(*&F(F(\"$D'F,F(%( ~.~.~.~GF(" }{XPPEDIT 18 0 "`` = Pi^4/90;" "6#/%!G*&%#PiG\"\"%\"#!*!\" \"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "sum(1/(n^4),n = 1 .. infinity);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$)%#PiG\"\"%\"\"\"#F(\"#!*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+MKK#3\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 4 " The " }{TEXT 261 21 "Riemann zeta function" }{TEXT -1 16 " is defined \+ for " }{XPPEDIT 18 0 "x > 1" "6#2\"\"\"%\"xG" }{TEXT -1 5 " by " } {XPPEDIT 18 0 "Zeta(x) = sum(1/(n^x),n = 1 .. infinity);" "6#/-%%ZetaG 6#%\"xG-%$sumG6$*&\"\"\"F,)%\"nGF'!\"\"/F.;F,%)infinityG" }{TEXT -1 47 ". The zeta function in Maple is designated by " }{TEXT 0 4 "Zeta " }{TEXT -1 2 ".\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "?Zeta" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Zeta(2);\nZeta(4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$)%#P iG\"\"#\"\"\"#F(\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$)%#PiG\" \"%\"\"\"#F(\"#!*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 64 "plot([Zeta(x),1],x=1..5,0..8,color=[red,blac k],linestyle=[1,4]);" }}{PARA 13 "" 1 "" {GLPLOT2D 361 267 267 {PLOTDATA 2 "6&-%'CURVESG6%7dp7$$\"+WYs-5!\"*$\"+%Ryfn$!\"(7$$\"+)G\\a +\"F*$\"+]#y3%=F-7$$\"+KR<35F*$\"+B&z\"H7F-7$$\"+x&)*3,\"F*$\"+h1JL#*! \")7$$\"+AKi85F*$\"+1\"G#)R(F=7$$\"+myM;5F*$\"+5F*$ \"+`'35I&F=7$$\"+arz@5F*$\"+^\\jXYF=7$$\"+)z@X-\"F*$\"+EU!f8%F=7$$\"+U kCF5F*$\"+)eB\"GPF=7$$\"+'3r*H5F*$\"+n%)[%R$F=7$$\"+JdpK5F*$\"+*Qik6$F =7$$\"+w.UN5F*$\"+'f77)GF=7$$\"+?]9Q5F*$\"+3GdzEF=7$$\"+k'p3/\"F*$\"+Z 4#[]#F=7$$\"+3VfV5F*$\"+Fb\">N#F=7$$\"+_*=j/\"F*$\"+\"H,q@#F=7$$\"+'fV !\\5F*$\"+4(zq4#F=7$$\"+S#o<0\"F*$\"+=Ny*)>F=7$$\"+&)G\\a5F*$\"+w)=K*= F=7$$\"+Iv@d5F*$\"+OF&e!=F=7$$\"+u@%*f5F*$\"+'yIks\"F=7$$\"+=omi5F*$\" +'z;Rl\"F=7$$\"+i9Rl5F*$\"+rsW(e\"F=7$$\"+1h6o5F*$\"+toHE:F=7$$\"+^2%3 2\"F*$\"+P=&)p9F=7$$\"+'RlN2\"F*$\"+l$*e<9F=7$$\"+S+Hw5F*$\"+S81p8F=7$ $\"+G$R<3\"F*$\"+,[r\"G\"F=7$$\"+<')=(3\"F*$\"+V9H07F=7$$\"+9+$>4\"F*$ \"+g3^7\"F*$\"+A0fy&)F*7 $$\"+z`3W6F*$\"+[_$z_(F*7$$\"+m40j6F*$\"+0)4?s'F*7$$\"+_(zV=\"F*$\"+4P 19gF*7$$\"+R&3d?\"F*$\"+\"eLKX&F*7$$\"+Et.F7F*$\"+sh0)*\\F*7$$\"+7hO[7 F*$\"+z5J@YF*7$$\"+ZkI\"H\"F*$\"+O&>3.%F*7$$\"+#yYUL\"F*$\"+ZE!Gf$F*7$ $\"+w#>(>9F*$\"+\"zY%*)HF*7$$\"+>K'*)\\\"F*$\"+K3Y;EF*7$$\"+Kd,\"e\"F* $\"+IW&*QBF*7$$\"+fX(em\"F*$\"+V]DD@F*7$$\"+U7Y]F*7$$\"+V! pu$=F*$\"+tirG=F*7$$\"+ib59>F*$\"+Se\\L$QF*$\"+ _1)[4\"F*7$$\"+qfa(>9F`hlF\\hl7$$\"3kmm;>K'*)\\\"F`hlF\\hl7$$\"3/++]Kd, \"e\"F`hlF\\hl7$$\"3gmm;fX(em\"F`hlF\\hl7$$\"3!*****\\U7Y]F`hlF\\hl7$$\"3#*******H,Q+?F`hl F\\hl7$$\"3)*******\\*3q3#F`hlF\\hl7$$\"3?+++q=\\q@F`hlF\\hl7$$\"3mmm; fBIYAF`hlF\\hl7$$\"30LLLj$[kL#F`hlF\\hl7$$\"3?LLL`Q\"GT#F`hlF\\hl7$$\" 3o****\\s]k,DF`hlF\\hl7$$\"3#HLLLvv-e#F`hlF\\hl7$$\"33++]sgamEF`hlF\\h l7$$\"3!)****\\$QF`hlF\\hl7$$\"3$*******pfa " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "The zeta function is defined for any co mplex number " }{XPPEDIT 18 0 "z = x +y*i" "6#/%\"zG,&%\"xG\"\"\"*&%\" yGF'%\"iGF'F'" }{TEXT -1 7 " with " }{XPPEDIT 18 0 "Re(z) > 1" "6#2 \"\"\"-%#ReG6#%\"zG" }{TEXT -1 21 " as the infinite sum " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Zeta(z) = sum(1/(n^z),n = 1 .. infinity);" "6#/-%%ZetaG6#%\"zG-%$sumG6$*&\"\"\"F,)%\"nGF'!\"\"/F.;F, %)infinityG" }{TEXT -1 3 " ." }}{PARA 0 "" 0 "" {TEXT -1 48 "It is al so possible to extend the definition of " }{XPPEDIT 18 0 "Zeta;" "6#%% ZetaG" }{TEXT -1 46 " to the whole of the complex plane except for " } {XPPEDIT 18 0 "z = 1" "6#/%\"zG\"\"\"" }{TEXT -1 202 " by a process ca lled analytic continuation.\n\nThe zeta function is at the heart of th e subject of analytic number theory, which ties together the fields of complex analysis and number theory. The famous " }{TEXT 261 18 "Riema nn Hypothesis" }{TEXT -1 237 " is a long-standing unsolved problem in \+ mathematics. It states that all the zeros of the zeta function, which \+ lie in the vertical strip consisting of complex numbers with their rea l part between 0 and 1, actually have real part equal to " }{XPPEDIT 18 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 45 ", that is, they all lie on the vertical line " }{XPPEDIT 18 0 "x = 1/2;" "6#/%\"xG*&\"\" \"F&\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 53 "We can \+ investigate the zeros along the critical line " }{XPPEDIT 18 0 "x = 1/ 2" "6#/%\"xG*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 60 " by constructing a re al valued function of a real variable " }{TEXT 264 1 "x" }{TEXT -1 17 " whose value is " }{XPPEDIT 18 0 "mzc(x) = abs(Zeta(1/2+x*I));" "6#/ -%$mzcG6#%\"xG-%$absG6#-%%ZetaG6#,&*&\"\"\"F0\"\"#!\"\"F0*&F'F0%\"IGF0 F0" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 48 "The following graph takes a few minutes to draw." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "mzc := x -> abs(Zeta(0.5+x*I ));\nplot('mzc'(x),x=0..50,numpoints=100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$mzcGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$absG6#-%%Z etaG6#,&$\"\"&!\"\"\"\"\"*&9$F6^#F6F6F6F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7e[l7$$\"\"!F)$\"+4XN g9!\"*7$$\"+#Hz?k#!#5$\"+/pe-8F,7$$\"+&eeTG&F0$\"+8ACR5F,7$$\"+]w,$e(F 0$\"+&)H2E')F07$$\"+;n(=))*F0$\"+N,M8uF07$$\"+;%=nC\"F,$\"+\"Gwr^'F07$ $\"+h\"\\_]\"F,$\"+j%=l%fF07$$\"+;O\\lS-$F,$\"+p^I'R&F07$$\"+*[jEF$F,$\"+LtOJbF07$$\"+puI@NF,$ \"+&[[]q&F07$$\"+)Q.c.%F,$\"+i5gshF07$$\"+a*\\#[XF,$\"+aB*zw'F07$$\"+f *pb2&F,$\"+rCh+vF07$$\"+2P.SbF,$\"+3.BO#)F07$$\"+y)4H1'F,$\"++J`^\"*F0 7$$\"+II$ze'F,$\"+%omV,\"F,7$$\"+@,*Q4(F,$\"+9II96F,7$$\"+JwM`vF,$\"+* pOf?\"F,7$$\"+]qo*4)F,$\"+ay\\68F,7$$\"+,#3Dc)F,$\"+g(*[$R\"F,7$$\"+E# z35*F,$\"+y8vt9F,7$$\"+&[_\"F,7 $$\"+>S&)Q)*F,$\"+`%\\Da\"F,7$$\"+Bz-55!\")$\"+-'[:b\"F,7$$\"+a[ZA5Fau $\"+Z+Y_:F,7$$\"+&y@\\.\"Fau$\"+9^.^:F,7$$\"+;(ot/\"Fau$\"+8y:Z:F,7$$ \"+Zc\")f5Fau$\"+QQrS:F,7$$\"+!y*y&3\"Fau$\"+`=^=:F,7$$\"+9Rw66Fau$\"+ u]n$[\"F,7$$\"+S'o%f6Fau$\"+\"*y(QQ\"F,7$$\"+JY#4@\"Fau$\"+e,.?7F,7$$ \"+XHPk7Fau$\"+Ra&z%)*F07$$\"+f)**3J\"Fau$\"+Y!4>E(F07$$\"+I]-O8Fau$\" +S,EncF07$$\"+,-:h8Fau$\"+TOLWRF07$$\"+hp5(Q\"Fau$\"+S8FU?F07$$\"+@P18 9Fau$\"+:`HTK!#77$$\"+ZtXQ9Fau$\"+aQ->?F07$$\"+t4&QY\"Fau$\"+q;rKTF07$ $\"+71U)[\"Fau$\"+R#\\O@'F07$$\"+]-*H^\"Fau$\"+$o%[/$)F07$$\"+W1FS:Fau $\"+vCVg5F,7$$\"+R5bn:Fau$\"+%GWUG\"F,7$$\"+(oj?f\"Fau$\"+])[Fk\"Fau$\"+r%QR$=F,7$$\"+.8#*o;Fau$\"+NFx *)>F,7$$\"+CON;Fau$\"+R8Mw=F,7$$\"+3*pR%>Fau$\"+`*)[v;F,7$$\"+,92q>Fau$\"+#owV W\"F,7$$\"+U-@&*>Fau$\"+,)os>\"F,7$$\"+$3\\.-#Fau$\"+wqu3$*F07$$\"+zr0 Y?Fau$\"+&[a$))F\\cl7$$\"+ 7dvA@Fau$\"+3=^/BF07$$\"+rJ=Y@Fau$\"+N*e$>[F07$$\"+I1hp@Fau$\"+(HO)erF 07$$\"+T6Y'>#Fau$\"+pp,^&*F07$$\"+`;JBAFau$\"+!orc:\"F,7$$\"+qqKZAFau$ \"+%*4G'H\"F,7$$\"+*[U8F#Fau$\"+&40bR\"F,7$$\"+5_9%G#Fau$\"+3f;I9F,7$$ \"+Kz%pH#Fau$\"+/_`^9F,7$$\"+a1v4BFau$\"+?VCf9F,7$$\"+xLbABFau$\"+6`,` 9F,7$$\"+a51ZBFau$\"+zBj,9F,7$$\"+K(o:P#Fau$\"+`F07$$\"+cx')*\\#Fa u$\"+F,7$$ \"+C67FHFau$\"+5aXj;F,7$$\"+Sh?SHFau$\"+VU'*z9F,7$$\"+c6H`HFau$\"+!>20 H\"F,7$$\"+shPmHFau$\"+\\r0(4\"F,7$$\"+*=h%zHFau$\"+,:b;!*F07$$\"+d@$> *HFau$\"+aL>brF07$$\"+EJS/IFau$\"+?(yLJ&F07$$\"+&4uo,$Fau$\"+%**[%4NF0 7$$\"+l]MHIFau$\"+84gh$F07$$\"+'\\X=3$Fau$\"+KMt'e%F07$$\"+i#3c5$F au$\"+a)p^q'F07$$\"+G5PHJFau$\"+p3PC#)F07$$\"+!=-D9$Fau$\"+%pB;y)F07$$ \"+LLjbJFau$\"+`?(p7*F07$$\"+&[k(oJFau$\"+42ba#*F07$$\"+Qc*==$Fau$\"+' [[4;*F07$$\"+].02KFau$\"+Qz$HO)F07$$\"+i]?KKFau$\"+$=hFx'F07$$\"+Z&)Hd KFau$\"+FBh_WF07$$\"+J?R#G$Fau$\"+R8[)[\"F07$$\"+#fW*)G$Fau$\"+$3XkA'F \\cl7$$\"+`r\\&H$Fau$\"+h'*GmFF\\cl7$$\"+8(\\?I$Fau$\"+f\"pn?\"F07$$\" +uAg3LFau$\"+%*\\4l@F07$$\"+&R2g`,\"F,7$$\"+*RB 5P$Fau$\"+%Q8O@\"F,7$$\"+%p$4$Q$Fau$\"+f>C59F,7$$\"+F`X&R$Fau$\"+0gt2; 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" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "mzc : = x -> abs(Zeta(0.5+x*I));\nevalf[15](fsolve(mzc(x),x=14));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$mzcGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-% $absG6#-%%ZetaG6#,&$\"\"&!\"\"\"\"\"*&9$F6^#F6F6F6F(F(F(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"0ZtT^sMT\"!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "We can also use the zeta function di rectly if the option " }{TEXT 0 7 "complex" }{TEXT -1 13 " is included ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "evalf[15](fsolve(Zeta(z),z=0.4+14*I,complex));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0+++++++&!#:$\"0ZtT^sMT\"!#8" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "Zeros of the zeta funct ion" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 262 9 "Ex ample 1" }{TEXT -1 17 ". The procedure " }{TEXT 0 10 "interproot" } {TEXT -1 33 " seems to work a bit faster than " }{TEXT 0 6 "fsolve" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "st := time():\nevalf[30](interproot(Zeta(z),z=0. 6+14*I..0.4+14*I));\ntime()-st;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$ \"?++++++++++++++]!#I$\"?O)>Dd/z$pM<9DZ89!#G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$G$!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "st := time():\nevalf[30](fsolve(Zet a(z),z=0.4+14*I..0.6+14.2*I,complex));\ntime()-st;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#^$$\"?++++++++++++++]!#I$\"?O)>Dd/z$pM<9DZ89!#G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$c'!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 262 9 "Example 2" }{TEXT -1 2 ".\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "st := time():\nevalf[30](interproot(Zeta(z) ,z=0.49+49.7*I..0.5+49.8*I));\ntime()-st;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"?++++++++++++++]!#I$\"?'yYy;>=-BnxC$Qx\\!#G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$1%!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "st := time():\neva lf[30](fsolve(Zeta(z),z=0.49+49.77*I..0.5+49.8*I,complex));\ntime()-st ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"?++++++++++++++]!#I$\"?'yYy; >=-BnxC$Qx\\!#G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%o9!\"$" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 262 9 "Example 3 " }{TEXT -1 2 ".\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "st := \+ time():\nevalf[30](interproot(Zeta(z),z=0.49+500.3*I..0.51+500.4*I)); \ntime()-st;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"?++++++++++++++]! #I$\"?D2R4$Rb\\!pT\\34.]!#F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%D6! \"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "st := time():\nevalf[30](fsolve(Zeta(z),z=0.49+500.3* I..0.51+500.4*I,complex));\ntime()-st;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"?++++++++++++++]!#I$\"?D2R4$Rb\\!pT\\34.]!#F" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"%4J!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1 " }}{PARA 0 "" 0 "" {TEXT -1 13 "The equation " }{XPPEDIT 18 0 "sin(z) = z" "6#/-%$sinG6#%\"zGF'" }{TEXT -1 29 " has a complex solution near " }{XPPEDIT 18 0 "14 + 3*i" "6#,&\"#9\"\"\"*&\"\"$F%%\"iGF%F%" } {TEXT -1 41 ". Find this solution correct to15 digits." }}{PARA 0 "" 0 "" {TEXT -1 47 "_______________________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 47 "____________ __________________________________ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }} {PARA 0 "" 0 "" {TEXT -1 90 "Check some of the zeros of the zeta funct ion listed above, which were not already checked." }}{PARA 0 "" 0 "" {TEXT -1 47 "_______________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 47 "______________________________________________ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }}{PARA 0 "" 0 "" {TEXT -1 68 "There are two zeros of \+ the zeta function on the critical line near 0" }{XPPEDIT 18 0 ".5 + 70 05*i" "6#,&-%&FloatG6$\"\"&!\"\"\"\"\"*&\"%0qF)%\"iGF)F)" }{TEXT -1 55 ", which are close together. \nThey are above and below 0" } {XPPEDIT 18 0 ".5 + 7005.1*i" "6#,&-%&FloatG6$\"\"&!\"\"\"\"\"*&-F%6$ \"&^+(F(F)%\"iGF)F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 18 "Th e one below is 0" }{XPPEDIT 18 0 ".5 + 7005.06286617492*i" "6#,&-%&Flo atG6$\"\"&!\"\"\"\"\"*&-F%6$\"0#\\ " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 47 "____________ __________________________________ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }