{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 "Purple Emphasis" -1 259 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 260 "Times " 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 261 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis " -1 262 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 0 1 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 "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {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 0 1 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 0 1 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 0 1 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 0 1 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 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" 260 277 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 278 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 279 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "Magenta Emphasis" -1 280 "Tim es" 1 12 255 0 255 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 "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 }{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 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 24 "The Lambert W functions " }} {PARA 0 "" 0 "" {TEXT -1 65 "by Peter Stone, Dept. of Applied and Envi ronmental Sciences, RMIT" }}{PARA 0 "" 0 "" {TEXT -1 61 "peter.stone@r mit.edu.au . . or . . peterstone@optusnet.com.au" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 11.12.2003" }} {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 30 "Solving equations of the form " } {XPPEDIT 18 0 "a^x = b*x+c" "6#/)%\"aG%\"xG,&*&%\"bG\"\"\"F&F*F*%\"cGF *" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 72 "For motivation we consider the problem of solving equations of the form " }{XPPEDIT 18 0 "a^x = b*x+c;" "6#/)% \"aG%\"xG,&*&%\"bG\"\"\"F&F*F*%\"cGF*" }{TEXT -1 47 ". Solutions can b e interpreted in terms of the " }{TEXT 263 1 "x" }{TEXT -1 68 " coordi nates of the points of intersection of the exponential graph " } {XPPEDIT 18 0 "y = a^x;" "6#/%\"yG)%\"aG%\"xG" }{TEXT -1 23 " with the linear graph " }{XPPEDIT 18 0 "y=b*x+c" "6#/%\"yG,&*&%\"bG\"\"\"%\"xG F(F(%\"cGF(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "As an example, consider the equation " } {XPPEDIT 18 0 "2^x=3*x-1" "6#/)\"\"#%\"xG,&*&\"\"$\"\"\"F&F*F*F*!\"\" " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 57 "It is easy to check \+ that this equation has the solutions " }{XPPEDIT 18 0 "x=1" "6#/%\"xG \"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x=3" "6#/%\"xG\"\"$" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 125 "However, it is not pos sible to give a step by step method using elementary mathematical func tions to arrive at these values. 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" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "2^x=3*x-1;\nsolve(%,x);\nop(map(expand@simplify@expand,[%]));\neva lf(evalf[14](%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)\"\"#%\"xG,&*& \"\"$\"\"\"F&F*F*F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&*&#\"\" \"\"\"$F&-%$expG6#,&-%)LambertWG6#,$*&#F&F'F&*&-%#lnG6#\"\"#F&)F6F%F&F &!\"\"F8*&F%F&F3F&F&F&F&F%F&,&*&F%F&-F)6#,&-F-6$F8F/F8*&F%F&F3F&F&F&F& F%F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&*&-%#lnG6#\"\"#!\"\"-%)Lambe rtWG6#,$*&#\"\"\"\"\"$F0*&F%F0)F(#F0F1F0F0F)F0F)F4F0,&*&F%F)-F+6$F)F-F 0F)F4F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+++++5!\"*$\"+++++IF%" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The equ ation " }{XPPEDIT 18 0 "3^x=5*x-2" "6#/)\"\"$%\"xG,&*&\"\"&\"\"\"F&F*F *\"\"#!\"\"" }{TEXT -1 18 " has the solution " }{XPPEDIT 18 0 "x=1" "6 #/%\"xG\"\"\"" }{TEXT -1 22 " and another solution " }{TEXT 265 1 "x" }{TEXT -1 1 " " }{TEXT 264 1 "~" }{TEXT -1 14 " 1.712494308. 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" }}{PARA 0 "" 0 "" {TEXT -1 90 "We use standar d graph-sketching techniques to determine the main features of the gra ph of " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 18 "The derivative of " }{XPPEDIT 18 0 "g(x); " "6#-%\"gG6#%\"xG" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "g*`'`(x)=exp(x) +x*exp(x)" "6#/*&%\"gG\"\"\"-%\"'G6#%\"xGF&,&-%$expG6#F*F&*&F*F&-F-6#F *F&F&" }{XPPEDIT 18 0 "``=(1+x)*exp(x)" "6#/%!G*&,&\"\"\"F'%\"xGF'F'-% $expG6#F(F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "g*`'`(x)=0" "6#/*&%\"gG\"\"\"-%\"'G6#%\"xGF&\"\"!" } {TEXT -1 6 " when " }{XPPEDIT 18 0 "x=-1" "6#/%\"xG,$\"\"\"!\"\"" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "g*`'`(x);" "6#*&%\"gG\"\"\"-%\"'G6#%\" xGF%" }{TEXT -1 18 " is negative when " }{XPPEDIT 18 0 "x<-1" "6#2%\"x G,$\"\"\"!\"\"" }{TEXT -1 19 " and positive when " }{XPPEDIT 18 0 "x>- 1" "6#2,$\"\"\"!\"\"%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 28 " is strictly decreasing for " }{XPPEDIT 18 0 "x<-1" "6#2%\"xG,$ \"\"\"!\"\"" }{TEXT -1 29 " and strictly increasing for " }{XPPEDIT 18 0 "x>-1" "6#2,$\"\"\"!\"\"%\"xG" }{TEXT -1 2 ". 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 25 "The second derivative of " }{XPPEDIT 18 0 "g(x)" "6# -%\"gG6#%\"xG" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "g*`''`(x)=exp(x)+exp (x)+x*exp(x)" "6#/*&%\"gG\"\"\"-%#''G6#%\"xGF&,(-%$expG6#F*F&-F-6#F*F& *&F*F&-F-6#F*F&F&" }{XPPEDIT 18 0 "``=(2+x)*exp(x)" "6#/%!G*&,&\"\"#\" \"\"%\"xGF(F(-%$expG6#F)F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g*`''`(x) = 0;" "6#/*&%\"gG\"\"\"-%#''G6#%\"x GF&\"\"!" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "x = -2;" "6#/%\"xG,$\" \"#!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "g*`''`(x);" "6#*&%\"gG\"\" \"-%#''G6#%\"xGF%" }{TEXT -1 18 " is negative when " }{XPPEDIT 18 0 "x < -2;" "6#2%\"xG,$\"\"#!\"\"" }{TEXT -1 19 " and positive when " } {XPPEDIT 18 0 "-2 < x;" "6#2,$\"\"#!\"\"%\"xG" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 26 "The graph of the function " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 27 " is concave downwards when " }{XPPEDIT 18 0 "x<-2" "6#2%\"xG,$\"\"#!\"\"" }{TEXT -1 25 " and concav e upwards for " }{XPPEDIT 18 0 "x>-2" "6#2,$\"\"#!\"\"%\"xG" }{TEXT -1 2 ". 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" }}{PARA 0 "" 0 "" {TEXT -1 3 "A s " }{XPPEDIT 18 0 "x->infinity" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6 \"%)infinityGF*F*F*" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "(x*exp(x))" "6#* &%\"xG\"\"\"-%$expG6#F$F%" }{TEXT -1 20 " tends to infinity. " }} {PARA 0 "" 0 "" {TEXT -1 3 "As " }{XPPEDIT 18 0 "x->-infinity" "6#f*6# %\"xG7\"6$%)operatorG%&arrowG6\",$%)infinityG!\"\"F*F*F*" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "x*exp(x)=x/exp(-x)" "6#/*&%\"xG\"\"\"-%$expG6#F%F &*&F%F&-F(6#,$F%!\"\"F." }{TEXT -1 13 " tends to 0. " }}{PARA 0 "" 0 " " {TEXT -1 41 "(This happens because, roughly speaking, " }{XPPEDIT 18 0 "exp(x)" "6#-%$expG6#%\"xG" }{TEXT -1 30 " grows much more rapidl y than " }{TEXT 266 1 "x" }{TEXT -1 48 ". More formally, L'Hospitals r ule can be used). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 276 "p1 := plot(x*exp(x),x=-1..1.4,color=red,thi ckness=2):\np2 := plot(x*exp(x),x=-4..-1,color=brown,thickness=2):\np3 := plot([[[-1,-exp(-1)]]$3],style=point,\n symbol=[circle,diamond,c ross],color=[black,green$2]):\nplots[display]([p1,p2,p3],view=[-4..1.5 ,-.6..4],labels=[`x`,`y`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 487 487 487 {PLOTDATA 2 "6)-%'CURVESG6%7S7$$!\"\"\"\"!$!3MBWr6WzyO!#=7$$!3!)****** *HooZ*F-$!3G2*fMa\"etOF-7$$!3T+++0Up@!*F-$!3iOJj[%)**fOF-7$$!3y******H L!)4&)F-$!3P_D`uIlLOF-7$$!3/+++5$>X*zF-$!3s7))Hw$QTf$F-7$$!3M+++XVo\"[ (F-$!3V[I60jdSNF-7$$!3t*****\\o?i+(F-$!3Cv@;3E-xMF-7$$!3/,++0c!R^'F-$! 3qudU<:$eR$F-7$$!3%******\\k_Z+'F-$!3![51=F7RH$F-7$$!3M+++XDB(\\&F-$!3 i'o'\\H^]sJF-7$$!3#*******Rd=v\\F-$!3NJ7%\\%**4DIF-7$$!3g+++ImO:XF-$!3 E$)[0JSquGF-7$$!3-+++?#>x*RF-$!3]Esh$fi.o#F-7$$!3/,+++j%zZ$F-$!3sWp0o$ yiX#F-7$$!31+++!y[q(HF-$!3/L5fu&>0@#F-7$$!3?,++Xe=ADF-$!3S`:#*>s#*f>F- 7$$!35+++?)48)>F-$!3v%zc4'R>D;F-7$$!3c+++!)o6B:F-$!3*=*o#*>=$zI\"F-7$$ !3A3++]cH,**!#>$!3q5B\\T,\"z'*)F`q7$$!3C?+++[X$=&F`q$!3H&*Qr&\\<;#\\F` q7$$!3i,P+++lNs!#A$!3$z+NEZE^B(F[r7$$\"3j,++]!\\<#\\F`q$\"3];/-F`/q^F` q7$$\"3'*)*****\\FY15F-$\"3(3([3aC.86F-7$$\"3))*****\\`R(y9F-$\"3!)p>F `9S9F-$\"3MG_Eb;YDCF-7$$\"3P*****\\:#H\\F-7$$\"3G)*****R3L*)RF-$\"3EyuDNg.XfF-7$$\"3Q)*****HY7#\\ %F-$\"3/VreRx]RqF-7$$\"3c)****\\Z.'y\\F-$\"3+(3E4D&y!>)F-7$$\"3?****** *Gb(=bF-$\"3GH`;ziN$e*F-7$$\"3)p*****>d5/gF-$\"3#))fXrloW4\"!#<7$$\"3? )*******3KAlF-$\"3-=aRN!p@D\"Fju7$$\"3o'****\\(3!>*pF-$\"3&>@.*Rg&oS\" Fju7$$\"3:******>eF0vF-$\"3s#o/BQ0(*e\"Fju7$$\"3Y(****\\j@$))zF-$\"3G& Rzwmedx\"Fju7$$\"3)*)****\\pVK\\)F-$\"3cxm`#)px&)>Fju7$$\"3K'******H(* o)*)F-$\"3!)y0R0d_2AFju7$$\"3c(****\\(oq.&*F-$\"3!***4\\cPHeCFju7$$\"3 (******>fX,+\"Fju$\"3BI)p]Vt!>FFju7$$\"3%******4iZ50\"Fju$\"3EgBX;cn1I Fju7$$\"30++]b\"G:5\"Fju$\"30pMlcUB9LFju7$$\"3%******R_9z9\"Fju$\"3Q!z A'QGymP(GFju$!3Z.]$z]pKi\"F-7$$!34+++&=$z9GFju$ !3aAP&oqUlo\"F-7$$!3;+]iX/4]FFju$!3UKnA+]\"zv\"F-7$$!39+](o8y%)o#Fju$! 3V;'*HuOuF=F-7$$!3\"****\\i:#>CEFju$!3%=WS*>Z\\->F-7$$!3o**\\7ev:lDFju $!3%z>SKM(ys>F-7$$!3.++vo2[,DFju$!39lFju$!3b=cnD-S2FF-7$$!3&*****\\())4Z$>Fju$!3QI* *4c8+&z#F-7$$!39+]i!R7g(=Fju$!3Ed*4![q/uGF-7$$!3$)****\\A0%=\"=Fju$!3s Yxgd&*pfHF-7$$!3?+]i&zf9v\"Fju$!3S*H*H!*49RIF-7$$!3'***\\7QXM)o\"Fju$! 3Szffyo[?JF-7$$!38++]PyjE;Fju$!3[oVN]!)z(>$F-7$$!39+]iSm.i:Fju$!3Q9B7c TsvKF-7$$!3#)******4!=)*\\\"Fju$!3wMp.@a:ZLF-7$$!3(****\\PZ!>O9Fju$!3p F-VE9s:MF-7$$!3#)**\\i0)*3t8Fju$!3%)4HS-FNyMF-7$$!3')*****\\%o5:8Fju$! 3_k;\">)pMINF-7$$!3#****\\(G=l[7Fju$!3Oq!\\(\\OF#e$F-7$$!3++++qO@*=\"F ju$!3jzl!\\z'o?OF-7$$!3$***\\i&>Se7\"Fju$!3UhbB\"H)*>l$F-7$$!3%***\\P% p$=l5Fju$!3a#QFzV58n$F-F'-F\\[l6&F^[l$\")#)eqkFa[l$\"))eqk\"Fa[lF^[mFc [l-F$6&7#F'-F\\[l6&F^[lF*F*F*-%'SYMBOLG6#%'CIRCLEG-%&STYLEG6#%&POINTG- F$6&Fb[m-F\\[l6&F^[lFb[lF_[lFb[l-Ff[m6#%(DIAMONDGFi[m-F$6&Fb[mF_\\m-Ff [m6#%&CROSSGFi[m-%+AXESLABELSG6%%\"xG%\"yG-%%FONTG6#%(DEFAULTG-%%VIEWG 6$;F[\\l$\"#:F);$!\"'F)$\"\"%F*" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 52 "Principal \+ (major) branch of the Lambert W function: " }{XPPEDIT 18 0 "W[0](x)" " 6#-&%\"WG6#\"\"!6#%\"xG" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 19 "Over the interval \+ \{" }{TEXT 269 1 "x" }{TEXT -1 1 " " }{TEXT 270 1 ":" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x in R" "6#-%#inG6$%\"xG%\"RG" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "-1<=x" "6#1,$\"\"\"!\"\"%\"xG" }{XPPEDIT 18 0 "`` " 0 "" {MPLTEXT 1 0 362 "p1 := plot(x*exp(x),x=- 1..1,color=red,thickness=2):\np2 := plot([y*exp(y),y,y=-1..2.7],color= blue,thickness=2):\np3 := plot(x,x=-1..2.7,color=black,linestyle=2,thi ckness=2):\np4 := plot([[[-1,-exp(-1)],[-exp(-1),-1]]$3],style=point, \n symbol=[circle,diamond,cross],color=[black,green$2]):\nplots[di splay]([p1,p2,p3,p4],view=[-1..2.7,-1..2.7],scaling=constrained);" }} {PARA 13 "" 1 "" {GLPLOT2D 477 477 477 {PLOTDATA 2 "6+-%'CURVESG6%7S7$ $!\"\"\"\"!$!3MBWr6WzyO!#=7$$!3ommm;p0k&*F-$!3umRSGa>vOF-7$$!3wKL$34%*Q?8i$F-7$$!3\"QLL3i.9!zF-$!3/K;5/w]&e$F-7$$!3\"ommT!R=0v F-$!3**yWs%HaF-$!3AmP27TqaJF-7$$!3Q+++]$*4)*\\F-$!3]ng 'Htw?.$F-7$$!39+++]_&\\c%F-$!3/Uo5S%)*=*GF-7$$!31+++]1aZTF-$!3#y)\\v.= YRFF-7$$!3umm;/#)[oPF-$!3W)[E@Wd_e#F-7$$!3hLLL$=exJ$F-$!3e9Dy\\J)4Q#F- 7$$!3*RLLLtIf$HF-$!3=8%4asq*)=#F-7$$!3]++]PYx\"\\#F-$!3U\\;;1H>U>F-7$$ !3EMLLL7i)4#F-$!3:*e\\%pQM,$!3%G5ZTC!f@wFdr7 $$!3KMLL3s$QM%Fdr$!3=Z/4nu=fTFdr7$$!3]^omm;zr)*!#@$!3Wzf:[70i)*F_s7$$ \"3%pJL$ezw5VFdr$\"3gImcI'e1]%Fdr7$$\"3s*)***\\PQ#\\\")Fdr$\"3[j?i*3\\ 6%))Fdr7$$\"3GKLLe\"*[H7F-$\"3q6Tv3\"R.R\"F-7$$\"3I*******pvxl\"F-$\"3 GI'=^7!pc>F-7$$\"3#z****\\_qn2#F-$\"3y\"eATf@hb#F-7$$\"3U)***\\i&p@[#F -$\"33xHu35\\\"=$F-7$$\"3B)****\\2'HKHF-$\"3Vg=HExZJRF-7$$\"3ElmmmZvOL F-$\"3aYW6$)*4%eYF-7$$\"3i******\\2goPF-$\"3i9$zKa!\\$\\&F-7$$\"3UKL$e R<*fTF-$\"3w&pqj!z!fI'F-7$$\"3m******\\)Hxe%F-$\"3gP#)F-7$$\"3y)***\\7k.6aF-$\"3/M\">f**GcH*F-7$$ \"3#emmmT9C#eF-$\"3\"3()RF4RA/\"!#<7$$\"33****\\i!*3`iF-$\"3?8&)zlFhw7$$\"3fKLLLbdQ()F-$\"3'H g1*z<)Q4#Fhw7$$\"3[++]i`1h\"*F-$\"3iKmVAX%)*G#Fhw7$$\"3W++]P?Wl&*F-$\" 3e(*=)Gu%e*[#Fhw7$$\"\"\"F*$\"34X!f%G=G=FFhw-%'COLOURG6&%$RGBG$\"*++++ \"!\")$F*F*Fb[l-%*THICKNESSG6#\"\"#-F$6%7U7$F+F(7$$!3=+mz3s;mOF-$!3)HL LeH0N>*F-7$$!3)4!G+l\")\\KOF-$!3qm;/m&y<\\)F-7$$!3Y%*)*\\Z(yac$F-$!3]L L3nMh-xF-7$$!3!Qj^3N]@Y$F-$!3:KLeps@3pF-7$$!3j./'3[d\"=LF-$!3Vn;a)p'f< hF-7$$!3:,:wU'*pUJF-$!3?M$3FA!f%Q&F-7$$!3lAU7P8g7HF-$!3A+]PuWgDYF-7$$! 3Eu\\m9<#eh#F-$!3)GL3xKg1%QF-7$$!3)H)o[!4SCD#F-$!3'***\\(o]L#eIF-7$$!3 %)[\"38yr()z\"F-$!3Imm;\\8T`AF-7$$!3`Jg?6!yMK\"F-$!3YKLeaN_W:F-7$$!36t e7m.*y#pFdr$!3!Q****\\(z$[Y(Fdr7$$\"3<)*))zxwU8b!#?$\"3!z4++vyK[&Fj_l7 $$\"3_m'p]4\"f$)*)Fdr$\"3%H,+](z\\q#)Fdr7$$\"3u>&G&e\"f1y\"F-$\"3(fm\" HAoHG:F-7$$\"3#[uZhiK:*HF-$\"3ALL$3OZ@O#F-7$$\"3;(=rZ7i0<%F-$\"3MLLLV \"G&oIF-7$$\"3Vw^b'Gd,u&F-$\"3O+]i?p@!*QF-7$$\"3)\\f')ey!RFtF-$\"3(>LL $=2bB%3he2$*F-$\"3Z,]iq6b:aF-7$$\"3IZy%yz_^9\"Fhw$\"3?-]( =JOa<'F-7$$\"3YjKT]4!))R\"Fhw$\"31NLeRnHopF-7$$\"3eoy#QUS;m\"Fhw$\"3#f Lek6!R'p(F-7$$\"3_\"o5c:13)>Fhw$\"3wmmT&=P<[)F-7$$\"39#R+,*p'eN#Fhw$\" 3_m;Hs?\\(H*F-7$$\"3@gALD4UAFFhw$\"36+v$*44w+5Fhw7$$\"33Yxm9$)pkJFhw$ \"3wm;H%\\bu2\"Fhw7$$\"3-TJUQ2cxOFhw$\"3%*****\\/&)oc6Fhw7$$\"3z(G5>ws -C%Fhw$\"3A+]7ZD?M7Fhw7$$\"3)>)fn&=a#[[Fhw$\"3&**\\i!p8?48Fhw7$$\"3'p^ A&o*HWg&Fhw$\"31+](QxuCR\"Fhw7$$\"3K5p;9;UkjFhw$\"3MLL$=j*Hn9Fhw7$$\"3 %eX.gK)4psFhw$\"39++vQ6>Z:Fhw7$$\"3w9!)QM*40=)Fhw$\"3]m\"H#=Ze>;Fhw7$$ \"3[_cM`q&pG*Fhw$\"3M++DA+t)p\"Fhw7$$\"3[&[m'RcNW5!#;$\"37LeR&e*>tnERK\"F^gl$\"3\\LL3nm 9F>Fhw7$$\"3UOF&QY,I\\\"F^gl$\"35+Dcw9#o+#Fhw7$$\"3clFs>:st;F^gl$\"3)o mmwydN3#Fhw7$$\"31t!R'R.ay=F^gl$\"3um;/\\<.i@Fhw7$$\"3Qn')e4xm.@F^gl$ \"3)Q$eR1f&)RAFhw7$$\"3#\\gk0pe?#e#Fhw7$$\"3M,c BM[!of$F^gl$\"3A+v$px1'>EFhw7$$\"3*oM11F%y,QF^gl$\"3?](o%)Q.)fEFhw7$$ \"32mcrlv_ " 0 "" {MPLTEXT 1 0 161 "a lias(W=LambertW):\ng := x -> x*exp(x):\n'g(x)'=g(x);\nxx := evalf(eval f[14](1/sqrt(2)));\n'W'(xx)=evalf(evalf[14](W(xx)));\n'g'(rhs(%))=eval f(evalf[14](g(rhs(%))));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6 #%\"xG*&F'\"\"\"-%$expGF&F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$ \"+7y1rq!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"WG6#$\"+7y1rq!#5$ \"+f^+1XF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#$\"+f^+1X!#5$\" +8y1rqF)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 103 ": So as to avoid displaying the assumed sy mbol ~, the following code use a double substitution \"trick\"." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "assume(x_>-1);\ng := x -> x*exp(x):\n'g(x)'=g(x);\n'W(g(x))'='W'( g(x));\n``=subs(x_=x,value(rhs(subs(x=x_,%))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG*&F'\"\"\"-%$expGF&F)" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"WG6#-%\"gG6#%\"xG-F%6#*&F*\"\"\"-%$expGF)F." }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G%\"xG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "g := x -> x*exp(x) :\n'g(x)'=g(x);\n'g(W(x))'=g(W(x));\n``=simplify(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG*&F'\"\"\"-%$expGF&F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#-%\"WG6#%\"xG*&F'\"\"\"-%$expGF&F, " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G%\"xG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 40 "Minor branch of the Lambert W function: " }{XPPEDIT 18 0 "W[-1](x);" "6#-&%\"WG6#,$\"\"\"!\"\"6#%\"xG" } {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 19 "Over the interval \{" }{TEXT 271 1 "x" }{TEXT -1 1 " " }{TEXT 272 1 ":" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x in R" "6#- %#inG6$%\"xG%\"RG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "-infinity < x;" "6 #2,$%)infinityG!\"\"%\"xG" }{XPPEDIT 18 0 "`` <= -1;" "6#1%!G,$\"\"\"! \"\"" }{TEXT -1 5 "\} = (" }{XPPEDIT 18 0 "-infinity,-1;" "6$,$%)infin ityG!\"\",$\"\"\"F%" }{TEXT -1 16 "], the function " }{XPPEDIT 18 0 "g (x)=x*exp(x)" "6#/-%\"gG6#%\"xG*&F'\"\"\"-%$expG6#F'F)" }{TEXT -1 6 " \+ is a " }{TEXT 259 19 "one-to-one function" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 41 "Consequently, it has an inverse function " } {XPPEDIT 18 0 "g^(-1)" "6#)%\"gG,$\"\"\"!\"\"" }{XPPEDIT 18 0 "``(x) = W[-1](x);" "6#/-%!G6#%\"xG-&%\"WG6#,$\"\"\"!\"\"6#F'" }{TEXT -1 15 ". The function " }{XPPEDIT 18 0 "W[-1](x);" "6#-&%\"WG6#,$\"\"\"!\"\"6# %\"xG" }{TEXT -1 8 " is the " }{TEXT 259 12 "minor branch" }{TEXT -1 8 " of the " }{TEXT 259 18 "Lambert W function" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "This func tion is available in Maple as " }{TEXT 262 14 "LambertW(-1,x)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 38 "In the following picture the \+ graph of " }{XPPEDIT 18 0 "g(x)=x*exp(x)" "6#/-%\"gG6#%\"xG*&F'\"\"\"- %$expG6#F'F)" }{TEXT -1 30 ", with the restricted domain \{" }{TEXT 273 1 "x" }{TEXT -1 1 " " }{TEXT 274 1 ":" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x in R" "6#-%#inG6$%\"xG%\"RG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 " -infinity < x;" "6#2,$%)infinityG!\"\"%\"xG" }{XPPEDIT 18 0 "`` <= -1; " "6#1%!G,$\"\"\"!\"\"" }{TEXT -1 5 "\} = (" }{XPPEDIT 18 0 "-infinity ,-1;" "6$,$%)infinityG!\"\",$\"\"\"F%" }{TEXT -1 15 "], is drawn in " }{TEXT 260 3 "red" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "The graph of " }{XPPEDIT 18 0 "g^(-1)" "6#)%\"gG,$\"\"\"!\"\"" }{XPPEDIT 18 0 "``(x) = W[-1](x);" "6#/-%!G6#%\"xG-&%\"WG6#,$\"\"\"!\"\"6#F'" } {TEXT -1 13 " is drawn in " }{TEXT 256 4 "blue" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 394 "alias(W=LambertW):\np1 := plot(x*exp(x),x=-3.3..-1,color=red,thic kness=2):\np2 := plot([y*exp(y),y,y=-3.3..-1],color=blue,thickness=2): \np3 := plot(x,x=-2.7..0.35,color=black,linestyle=2,thickness=2):\np4 \+ := plot([[[-1,-exp(-1)],[-exp(-1),-1]]$3],style=point,\n symbol=[c ircle,diamond,cross],color=[black,green$2]):\nplots[display]([p1,p2,p3 ,p4],view=[-3.3..0.35,-3.3..0.35],scaling=constrained);" }}{PARA 13 " " 1 "" {GLPLOT2D 477 477 477 {PLOTDATA 2 "6+-%'CURVESG6%7S7$$!3#)***** ********H$!#<$!3S?4CCX9<7!#=7$$!3UmmTXl')\\KF*$!3Gp9D(=y-E\"F-7$$!3JLe k%pXi?$F*$!3_X&>B7*z)H\"F-7$$!3qm;z&[*=dJF*$!31OC414BV8F-7$$!3Xm;a+\"3 y5$F*$!3w)[?Nl`\"*Q\"F-7$$!3ALeR;9meIF*$!3LBk()Qo0O9F-7$$!3Lm\"z*[h48I F*$!3%H?=yO91[\"F-7$$!3'**\\7Q&f\"f'HF*$!3=hR76,%y_\"F-7$$!3Tm\"z%G@7< HF*$!3_&RJo$e%yd\"F-7$$!3'**\\iby%[oGF*$!3a]^N&Hm)G;F-7$$!3+LL3+`X=GF* $!3i,_T,Rc#o\"F-7$$!3cm;/-$*QuFF*$!3'4ZJ'HR(3t\"F-7$$!3y***\\_U\"yCFF* $!3;gkwr6Q'y\"F-7$$!3'****\\P&)p\\n#F*$!3HP+DaCHV=F-7$$!3')***\\ZF-7$$!37LL3\">U:`#F*$ !3dPDB>T\\8?F-7$$!39LLLM?j([#F*$!3agGI#f%Qn?F-7$$!37+DJ$3alV#F*$!3-6aN v?0J@F-7$$!3=LL$=WT8R#F*$!3qi^`N7C)=#F-7$$!31+DJ3gtTBF*$!3)*HX)eSE=D#F -7$$!35+v$\\!**\\%H#F*$!3\"=KFncEJJ#F-7$$!3QL$ek)R@XAF*$!3)p4UC3\"zxBF -7$$!3WLe*y7a**>#F*$!3XdF\\x/tPCF-7$$!3cm;/c_8^@F*$!3V+6^_())G]#F-7$$! 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3[oVN]!)z(>$F*$!38++]PyjE;Fav7$$!3Q9B7cTsvKF*$!39+]iSm.i:Fav7$$!3wMp.@ a:ZLF*$!3#)******4!=)*\\\"Fav7$$!3pF-VE9s:MF*$!3(****\\PZ!>O9Fav7$$!3% )4HS-FNyMF*$!3#)**\\i0)*3t8Fav7$$!3_k;\">)pMINF*$!3')*****\\%o5:8Fav7$ $!3Oq!\\(\\OF#e$F*$!3#****\\(G=l[7Fav7$$!3jzl!\\z'o?OF*$!3++++qO@*=\"F av7$$!3UhbB\"H)*>l$F*$!3$***\\i&>Se7\"Fav7$$!3a#QFzV58n$F*$!3%***\\P%p $=l5FavF'-F[[l6&F][lFa[lFa[lF^[l-F$6&7#F'-F[[l6&F][lF-F-F--%'SYMBOLG6# %'CIRCLEG-%&STYLEG6#%&POINTG-F$6&Fijl-F[[l6&F][lFa[lF^[lFa[l-F][m6#%(D IAMONDGF`[m-F$6&FijlFf[m-F][m6#%&CROSSGF`[m-%%TEXTG6&7$$\"$m\"!\"#$\"$ =\"Ff\\mQ+y~=~W~~(x)6\"-%&COLORG6&F][l$\"#&*Ff\\mF-F--%%FONTG6$%*HELVE TICAG\"#5-Fa\\m6&7$$\"$x\"Ff\\m$\"#6F,Q\"0Fj\\mF[]m-Fa]m6$Fc]m\"\")-Fa \\m6&7$$F`[lF,$!#AF,Fi\\m-F\\]m6&F][lF-F-F^]mF`]m-Fa\\m6&7$$!#oFf\\m$! $G#Ff\\mQ#-1Fj\\mFf^mF]^m-Fa\\m6&7$$\"#VF,$F,F,Q\"xFj\\m-F\\]mF[[mF`]m -Fa\\m6&7$$!#8Ff\\m$\"$F\"Ff\\mQ\"yFj\\mFg_mF`]m-%+AXESLABELSG6%Q!Fj\\ mFc`m-Fa]m6#%(DEFAULTG-Fa]m6$Fc]m\"\"*-%%VIEWG6$Ff`mFf`m" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 44.000000 0 0 "Curve 1" "Curve 2" "Curv e 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curv e 10" "Curve 11" }}{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 43 "Formulas involving the Lambert W function s " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Give n " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=W[0](x)" "6#/ %\"yG-&%\"WG6#\"\"!6#%\"xG" }{TEXT -1 6 " or " }{XPPEDIT 18 0 "y=W[- 1](x)" "6#/%\"yG-&%\"WG6#,$\"\"\"!\"\"6#%\"xG" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 8 "we have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x=y*exp(y)" "6#/%\"xG*&%\"yG\"\"\"-%$expG6#F&F'" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dy=exp(y)+y*exp(y)" "6#/*&%#d xG\"\"\"%#dyG!\"\",&-%$expG6#%\"yGF&*&F-F&-F+6#F-F&F&" }{TEXT -1 1 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=x/y+x" "6#/%!G ,&*&%\"xG\"\"\"%\"yG!\"\"F(F'F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (y+1)*x/y;" "6#/%!G*(,&%\"yG\"\"\" F(F(F(%\"xGF(F'!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "H ence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = y/(( 1+y)*x);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&%\"yGF&*&,&F&F&F*F&F&%\"xGF&F( " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 40 "Thus we have the fol lowing derivatives. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[g(x), `|`, g*`'`(x)], [__ ______, __, ______________], [W[0](x), `|`, W[0](x)/((1+W[0](x))*x)], \+ [W[-1](x), `|`, W[-1](x)/((1+W[-1](x))*x)]]);" "6#-%'matrixG6#7&7%-%\" gG6#%\"xG%\"|grG*&F)\"\"\"-%\"'G6#F+F.7%%)________G%#__G%/____________ __G7%-&%\"WG6#\"\"!6#F+F,*&-&F96#F;6#F+F.*&,&F.F.-&F96#F;6#F+F.F.F+F.! \"\"7%-&F96#,$F.FH6#F+F,*&-&F96#,$F.FH6#F+F.*&,&F.F.-&F96#,$F.FH6#F+F. F.F+F.FH" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 278 21 "_____________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "The derivatives of the fu nctions " }{XPPEDIT 18 0 "W[0](x)" "6#-&%\"WG6#\"\"!6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "W[-1](x)" "6#-&%\"WG6#,$\"\"\"!\"\"6#%\"x G" }{TEXT -1 20 " do not exist where " }{XPPEDIT 18 0 "x=-1/exp(1)" "6 #/%\"xG,$*&\"\"\"F'-%$expG6#F'!\"\"F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "In the following pic ture the graph of " }{XPPEDIT 18 0 "W[0](x);" "6#-&%\"WG6#\"\"!6#%\"xG " }{TEXT -1 15 " is plotted in " }{TEXT 260 3 "red" }{TEXT -1 36 ", wh ile the graph of the derivative " }{XPPEDIT 18 0 "Diff(W[0](x),x) = W[ 0](x)/((1+W[0](x))*x)" "6#/-%%DiffG6$-&%\"WG6#\"\"!6#%\"xGF-*&-&F)6#F+ 6#F-\"\"\"*&,&F3F3-&F)6#F+6#F-F3F3F-F3!\"\"" }{TEXT -1 15 " is plotted in " }{TEXT 256 4 "blue" }{TEXT -1 2 ". 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" }}{PARA 0 "" 0 "" {TEXT -1 8 "Letting " }{XPPEDIT 18 0 "u=x+c" "6#/%\"uG,&%\"xG\"\"\"%\"cGF'" }{TEXT -1 24 ", the equation \+ becomes: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(u-c) =u" "6#/-%$expG6#,&%\"uG\"\"\"%\"cG!\"\"F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "exp(u)*exp(-c)=u" "6#/*&-%$expG6#%\"uG\"\"\"-F&6#,$%\"c G!\"\"F)F(" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 31 "This equa tion is equivalent to " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-u*exp(-u)=-exp(-c)" "6#/,$*&%\"uG\"\"\"-%$expG6#,$F&!\"\"F'F,,$ -F)6#,$%\"cGF,F," }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so th at " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-u = W[0](-1/e xp(c));" "6#/,$%\"uG!\"\"-&%\"WG6#\"\"!6#,$*&\"\"\"F/-%$expG6#%\"cGF&F &" }{TEXT -1 6 " or " }{XPPEDIT 18 0 "-u = W[-1](-1/exp(c));" "6#/,$ %\"uG!\"\"-&%\"WG6#,$\"\"\"F&6#,$*&F,F,-%$expG6#%\"cGF&F&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-x-c = W[0](-1/exp(c));" "6#/,&%\"xG!\" \"%\"cGF&-&%\"WG6#\"\"!6#,$*&\"\"\"F0-%$expG6#F'F&F&" }{TEXT -1 6 " o r " }{XPPEDIT 18 0 "-x-c = W[-1](-1/exp(c));" "6#/,&%\"xG!\"\"%\"cGF& -&%\"WG6#,$\"\"\"F&6#,$*&F-F--%$expG6#F'F&F&" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = -W[0](-1/exp(c))-c;" "6#/%\"xG,&-&%\"WG6#\"\"!6 #,$*&\"\"\"F.-%$expG6#%\"cG!\"\"F3F3F2F3" }{TEXT -1 6 " or " } {XPPEDIT 18 0 "x = -W[-1](-1/exp(c))-c;" "6#/%\"xG,&-&%\"WG6#,$\"\"\"! \"\"6#,$*&F+F+-%$expG6#%\"cGF,F,F,F3F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The solutions of the equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(x)= x+c" "6#/-%$expG6#%\"xG,&F'\"\"\"%\"cGF)" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 26 "can be interpreted as the " }{TEXT 275 1 "x" } {TEXT -1 57 " coordinates of the points of intersection of the graphs \+ " }{XPPEDIT 18 0 "y=x+c" "6#/%\"yG,&%\"xG\"\"\"%\"cGF'" }{TEXT -1 5 " \+ and " }{XPPEDIT 18 0 "y=exp(x)" "6#/%\"yG-%$expG6#%\"xG" }{TEXT -1 2 " . 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" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 103 ": So as to avoid displayin g the assumed symbol ~, the following code use a double substitution \+ \"trick\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 65 "assume(c_>1):\nexp(x)=x+c;\nop(subs(c_=c,[solve(sub s(c=c_,%),x)]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$expG6#%\"xG,&F '\"\"\"%\"cGF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&-%\"WG6#,$*&\"\"\" F)-%$expG6#%\"cG!\"\"F.F.F-F.,&-F%6$F.F'F.F-F." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 47 "For a numerical example, consider the equation " } {XPPEDIT 18 0 "exp(x)=x+2" "6#/-%$expG6#%\"xG,&F'\"\"\"\"\"#F)" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 18 "The solutions are " } {XPPEDIT 18 0 "x=-W[0](-1/exp(2))-2" "6#/%\"xG,&-&%\"WG6#\"\"!6#,$*&\" \"\"F.-%$expG6#\"\"#!\"\"F3F3F2F3" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x=-W[-1](-1/exp(2))-2" "6#/%\"xG,&-&%\"WG6#,$\"\"\"!\"\"6#,$*&F+F+-%$ expG6#\"\"#F,F,F,F3F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "exp(x)=x+2;\nsolve(%,x); \nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$expG6#%\"xG,&F'\"\" \"\"\"#F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&-%\"WG6#,$-%$expG6#!\"# !\"\"F,\"\"#F,,&-F%6$F,F'F,F-F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$!+ gcST=!\"*$\"+@K>Y6F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "The solutions are illustrated by the x coordinates of \+ the points of intersection of the two graphs " }{XPPEDIT 18 0 "y=x+2" "6#/%\"yG,&%\"xG\"\"\"\"\"#F'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y=e xp(x)" "6#/%\"yG-%$expG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "plot([exp(x) ,x+2],x=-2.6..1.7,y,color=[red,blue]);" }}{PARA 13 "" 1 "" {GLPLOT2D 336 326 326 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$!33+++++++E!#<$\"3p(QL9#yN Fu!#>7$$!3ZLL3PAF1DF*$\"3M=\\AL`F*$\"390/Iqp/(Q\"FB7$$!3[Le*)[[=%)=F*$\"3^gg6N(G&>:FB7$$ !3=+DJkTD$z\"F*$\"397l-ptFB7$$!33++Dg8fC:F*$\"3wN'=l.+r<#FB7$$!3-++vy` YJ9F*$\"3+K:Z3]e*Q#FB7$$!3,++vR7sT8F*$\"3ynTlqN&Rh#FB7$$!3QLe*Q'\\Ag7F *$\"3a/0i\"F*7$$\"3?JLL[F-uEFB$\"3oCFARfc18F*7$$\"3h++]7;\\-OFB$\"3OO&FB$\"32&>Y0Bv (4LLe*4>=!)FB$\"3ozmf$4$fHAF*7$$\"3*))*\\P%[TT%*)FB$\"3Q%*Q(QV -fW#F*7$$\"34qmmc=%f$)*FB$\"3a+4&H-]Sn#F*7$$\"3um;HaOzu5F*$\"3bljWr$)Q HHF*7$$\"3YL3FXzBl6F*$\"3s@^:3eo1KF*7$$\"3O++]boM[7F*$\"3yIv#HoxX[$F*7 $$\"3IL$ea/*fV8F*$\"3wp&H0=8G$QF*7$$\"3ymmmRPzG9F*$\"3iH%e:F*$\"3'Q2rhXH0d%F*7$$\"3()*\\i!Q+d1;F*$\"31I%e-<\"o&)\\F* 7$$\"3%**************p\"F*$\"3-*>F$!3LLL3(4ZoS#FB7$FD$!3Ym;z%yfCF-7$FS$\"39l;/6::e6FB7$FX$\"3C) *\\(oNeu1#FB7$Fgn$\"3+KL$e@#z-IFB7$F\\o$\"3Lnm\"zyMm#QFB7$Fao$\"3=**** \\(R'3aZFB7$Ffo$\"3%)****\\7iM&o&FB7$F[p$\"3%)****\\-wy#e'FB7$F`p$\"3; m;/h.v(R(FB7$Fep$\"3oLL$e!*>oO)FB7$Fjp$\"3/KLLB*[x=*FB7$F_q$\"3!**\\PH XoU,\"F*7$Fdq$\"3ALL$[V'z)4\"F*7$Fiq$\"3#**\\Pz-P:>\"F*7$F^r$\"3)**\\7 Q+[)z7F*7$Fcr$\"3QL$3FC\"*>P\"F*7$Fhr$\"3iL3-+vgc9F*7$F]s$\"3]m;zkv(ya \"F*7$Fbs$\"3gmT56:oU;F*7$Fgs$\"3$)*\\i]i3_s\"F*7$F\\t$\"3^m;/p,M9=F*7 $Fat$\"3p****\\vF*7$Fft$\"3w**\\(Gm0l*>F*7$F\\u$\"3y*\\PfXmO3#F*7$ Fau$\"3-+]7cOW!=#F*7$Ffu$\"37LL$[F-uE#F*7$F[v$\"31++Dh\"\\-O#F*7$F`v$ \"3QmT5SAQWCF*7$Fev$\"3J++v<>OODF*7$Fjv$\"32L3FEw!Hi#F*7$F_w$\"32+voGG P8FF*7$Fdw$\"3>LLe*4>=!GF*7$Fiw$\"3))*\\P%[TT%*GF*7$F^x$\"3-nmm&=%f$)H F*7$Fcx$\"3um;HaOzuIF*7$Fhx$\"3YL3FXzBlJF*7$F]y$\"3O++]boM[KF*7$Fby$\" 3IL$ea/*fVLF*7$Fgy$\"3ymmmRPzGMF*7$F\\z$\"3G+v$H0H'>NF*7$Faz$\"3()*\\i !Q+d1OF*7$Ffz$\"3;+++++++PF*-F[[l6&F][lFa[lFa[lF^[l-%+AXESLABELSG6$Q\" x6\"Q\"yF_el-%%VIEWG6$;$!#E!\"\"$\"# " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "Solution of " }{XPPEDIT 18 0 "a^x = b*x+c;" "6#/)%\"aG%\"xG,&*&%\"bG\"\"\"F&F*F*%\"cGF*" }{TEXT -1 31 " using the Lambert W functions " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 13 "The equation " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a^x=b*x+c" "6#/)%\"aG %\"xG,&*&%\"bG\"\"\"F&F*F*%\"cGF*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 27 "can be written in the form " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(x*ln*a)=b*x+c" "6#/-%$expG6#*(%\"xG\"\"\" %#lnGF)%\"aGF),&*&%\"bGF)F(F)F)%\"cGF)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "Suppose that " }{XPPEDIT 18 0 "b<>0" "6#0%\"bG\"\"! " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 8 "Letting " }{XPPEDIT 18 0 "u=b*x+c" "6#/%\"uG,&*&%\"bG\"\"\"%\"xGF(F(%\"cGF(" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "x=(u-c)/b" "6#/%\"xG*&,&%\"uG\"\"\"%\"cG! \"\"F(%\"bGF*" }{TEXT -1 25 ", we obtain the equation " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp((u-c)/b*ln*a)=u" "6#/-%$expG 6#**,&%\"uG\"\"\"%\"cG!\"\"F*%\"bGF,%#lnGF*%\"aGF*F)" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 33 "This equation is equivalent to: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(u*ln*a/b)*exp(-c* ln*a/b)=u" "6#/*&-%$expG6#**%\"uG\"\"\"%#lnGF*%\"aGF*%\"bG!\"\"F*-F&6# ,$**%\"cGF*F+F*F,F*F-F.F.F*F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-u*ex p(-u*ln*a/b)=-exp(-c*ln*a/b)" "6#/,$*&%\"uG\"\"\"-%$expG6#,$**F&F'%#ln GF'%\"aGF'%\"bG!\"\"F0F'F0,$-F)6#,$**%\"cGF'F-F'F.F'F/F0F0F0" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-``(u*ln*a/b)*exp(-u*ln*a/b) = -``(ln*a /b)*exp(-c*ln*a/b);" "6#/,$*&-%!G6#**%\"uG\"\"\"%#lnGF+%\"aGF+%\"bG!\" \"F+-%$expG6#,$**F*F+F,F+F-F+F.F/F/F+F/,$*&-F'6#*(F,F+F-F+F.F/F+-F16#, $**%\"cGF+F,F+F-F+F.F/F/F+F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-``(u*ln*a/b) = W[0](-``(ln*a/b)*exp(-c*ln*a/b));" "6#/,$-%!G6#**%\"u G\"\"\"%#lnGF*%\"aGF*%\"bG!\"\"F.-&%\"WG6#\"\"!6#,$*&-F&6#*(F+F*F,F*F- F.F*-%$expG6#,$**%\"cGF*F+F*F,F*F-F.F.F*F." }{TEXT -1 8 " or " } {XPPEDIT 18 0 "-``(u*ln*a/b) = W[-1](-``(ln*a/b)*exp(-c*ln*a/b));" "6# /,$-%!G6#**%\"uG\"\"\"%#lnGF*%\"aGF*%\"bG!\"\"F.-&%\"WG6#,$F*F.6#,$*&- F&6#*(F+F*F,F*F-F.F*-%$expG6#,$**%\"cGF*F+F*F,F*F-F.F.F*F." }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u = -``(b/(ln*a))*W[0](-``(ln*a/b)*exp( -c*ln*a/b));" "6#/%\"uG,$*&-%!G6#*&%\"bG\"\"\"*&%#lnGF,%\"aGF,!\"\"F,- &%\"WG6#\"\"!6#,$*&-F(6#*(F.F,F/F,F+F0F,-%$expG6#,$**%\"cGF,F.F,F/F,F+ F0F0F,F0F,F0" }{TEXT -1 8 " or " }{XPPEDIT 18 0 "u = -``(b/(ln*a)) *W[-1](-``(ln*a/b)*exp(-c*ln*a/b));" "6#/%\"uG,$*&-%!G6#*&%\"bG\"\"\"* &%#lnGF,%\"aGF,!\"\"F,-&%\"WG6#,$F,F06#,$*&-F(6#*(F.F,F/F,F+F0F,-%$exp G6#,$**%\"cGF,F.F,F/F,F+F0F0F,F0F,F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "exp(-c*ln*a/b)=a^(-c/b)" "6#/-%$expG6#,$**%\"cG\"\"\"%#lnGF*%\"aGF*% \"bG!\"\"F.)F,,$*&F)F*F-F.F." }{TEXT -1 5 " and " }{XPPEDIT 18 0 "u=b* x+c" "6#/%\"uG,&*&%\"bG\"\"\"%\"xGF(F(%\"cGF(" }{TEXT -1 12 ", we obta in " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b*x+c = -``(b/ (ln*a))*W[0](-``(ln*a/b)*a^(-c/b));" "6#/,&*&%\"bG\"\"\"%\"xGF'F'%\"cG F',$*&-%!G6#*&F&F'*&%#lnGF'%\"aGF'!\"\"F'-&%\"WG6#\"\"!6#,$*&-F-6#*(F1 F'F2F'F&F3F')F2,$*&F)F'F&F3F3F'F3F'F3" }{TEXT -1 8 " or " } {XPPEDIT 18 0 "b*x+c = -``(b/(ln*a))*W[-1](-``(ln*a/b)*a^(-c/b));" "6# /,&*&%\"bG\"\"\"%\"xGF'F'%\"cGF',$*&-%!G6#*&F&F'*&%#lnGF'%\"aGF'!\"\"F '-&%\"WG6#,$F'F36#,$*&-F-6#*(F1F'F2F'F&F3F')F2,$*&F)F'F&F3F3F'F3F'F3" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "This gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "x = -``(1/(ln*a))*W[0](-``(ln*a/b)*a^(-c/b))-c/b;" "6#/ %\"xG,&*&-%!G6#*&\"\"\"F+*&%#lnGF+%\"aGF+!\"\"F+-&%\"WG6#\"\"!6#,$*&-F (6#*(F-F+F.F+%\"bGF/F+)F.,$*&%\"cGF+F;F/F/F+F/F+F/*&F?F+F;F/F/" } {TEXT -1 8 " or " }{XPPEDIT 18 0 "x = -``(1/(ln*a))*W[-1](-``(ln*a /b)*a^(-c/b))-c/b;" "6#/%\"xG,&*&-%!G6#*&\"\"\"F+*&%#lnGF+%\"aGF+!\"\" F+-&%\"WG6#,$F+F/6#,$*&-F(6#*(F-F+F.F+%\"bGF/F+)F.,$*&%\"cGF+F;F/F/F+F /F+F/*&F?F+F;F/F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "Delta=``(b/(ln*a))*a ^(c/b)" "6#/%&DeltaG*&-%!G6#*&%\"bG\"\"\"*&%#lnGF+%\"aGF+!\"\"F+)F.*&% \"cGF+F*F/F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 13 "Suppose that " }{XPPEDIT 18 0 "0 < b/(ln* a);" "6#2\"\"!*&%\"bG\"\"\"*&%#lnGF'%\"aGF'!\"\"" }{TEXT -1 10 ", so t hat " }{XPPEDIT 18 0 "Delta>0" "6#2\"\"!%&DeltaG" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "When " } {XPPEDIT 18 0 "Deltaexp(1)" "6#2-%$expG6#\"\"\"%&DeltaG" }{TEXT -1 54 ", there will be two distinct real solutions given by: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = -W[0](-1/Delta)/( ln*a)-c/b" "6#/%\"xG,&*&-&%\"WG6#\"\"!6#,$*&\"\"\"F/%&DeltaG!\"\"F1F/* &%#lnGF/%\"aGF/F1F1*&%\"cGF/%\"bGF1F1" }{TEXT -1 8 " or " } {XPPEDIT 18 0 "x = -W[-1](-1/Delta)/(ln*a)-c/b" "6#/%\"xG,&*&-&%\"WG6# ,$\"\"\"!\"\"6#,$*&F,F,%&DeltaGF-F-F,*&%#lnGF,%\"aGF,F-F-*&%\"cGF,%\"b GF-F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "Suppose that " } {XPPEDIT 18 0 "b/(ln*a) < 0;" "6#2*&%\"bG\"\"\"*&%#lnGF&%\"aGF&!\"\"\" \"!" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "Delta<0" "6#2%&DeltaG\" \"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 42 "There will be one real solution given by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "x = -W[0](-1/Delta)/(ln*a)-c/b" "6#/%\"xG,&*&-&%\"WG6# \"\"!6#,$*&\"\"\"F/%&DeltaG!\"\"F1F/*&%#lnGF/%\"aGF/F1F1*&%\"cGF/%\"bG F1F1" }{TEXT -1 2 ". " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The equation " } {XPPEDIT 18 0 "3^x = 5*x-2" "6#/)\"\"$%\"xG,&*&\"\"&\"\"\"F&F*F*\"\"#! \"\"" }{TEXT -1 134 ", which was given as an example in the first sect ion, can be solved with the aid of the Lambert W functions using the f ormulas above. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 183 "a := 3;\nb := 5;\nc := -2;\nDelta := b/ln(a)* a^(c/b);\nevalf(%);\nx1 := -W(0,-1/Delta)/ln(a)-c/b;\n``=evalf(%);\nx2 := -W(-1,-1/Delta)/ln(a)-c/b;\n``=evalf(%);\nunassign('Delta','a','b' ,'c'):\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"\"$" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"bG\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"cG!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&DeltaG,$*&#\"\"&\"\"$ \"\"\"*&-%#lnG6#F)!\"\"F)#F)F(F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# $\"+\\NwKH!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G,&*&-%\"WG6#,$ *&#\"\"\"\"\"&F-*&-%#lnG6#\"\"$F-)F3#\"\"#F.F-F-!\"\"F-F0F7F7F5F-" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"+++++5!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x2G,&*&-%\"WG6$!\"\",$*&#\"\"\"\"\"&F.*&-%#lnG6#\"\" $F.)F4#\"\"#F/F.F.F*F.F1F*F*F6F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% !G$\"+2V\\7 \+ " 0 "" {MPLTEXT 1 0 48 "3^x=5*x-2;\nop(simplify([solve(%,x)]));\nevalf (%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)\"\"$%\"xG,&*&\"\"&\"\"\"F&F *F*\"\"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,$*&#\"\"\"\"\"&F&*&, &*&F'F&-%\"WG6#,$*&#F&F'F&*&-%#lnG6#\"\"$F&)F5#\"\"#F'F&F&!\"\"F&F9*&F 8F&F2F&F&F&F2F9F&F&,$*&F%F&*&,&*&F'F&-F,6$F9F.F&F9*&F8F&F2F&F&F&F2F9F& F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+++++5!\"*$\"+2V\\7 " 0 "" {MPLTEXT 1 0 55 "plot([e xp(x),2-x],x=-2..2.5,y=-.5..5,color=[red,blue]);" }}{PARA 13 "" 1 "" {GLPLOT2D 323 352 352 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$!\"#\"\"!$\"3-Fh OKGN`8!#=7$$!3#****\\i!G\">!>!#<$\"3w-^\")oz#G\\\"F-7$$!34+vVjwc;=F1$ \"3Ff7VL$Gei\"F-7$$!3')**\\PC\")e?F-7$$!30+vo9e\"y_\"F1$\"3kR1E#Q\"4q@F-7$$!3++vVyjmQ9F 1$\"3iF1P9%RCP#F-7$$!31+v$4IdjM\"F1$\"3b!*R`pJ'=g#F-7$$!3++v$47\"*3D\" F1$\"3OHYf,g\\iGF-7$$!33+v=-6tb6F1$\"3'=X#H=GF[JF-7$$!3))***\\iKZy0\"F 1$\"3)okO(*>C?Z$F-7$$!3W++DJCJ;(*F-$\"3wM%3fl^Yy$F-7$$!3k)***\\PNsX()F -$\"3sGc\"GF./<%F-7$$!3%3++DJ\\6x(F-$\"3?nR)z`/tf%F-7$$!3t)***\\ik'>$o F-$\"3mHoXF5+]]F-7$$!3s+]Pf%)4zfF-$\"33l>4m$*f*\\&F-7$$!3!)****\\7f&\\ '\\F-$\"3;lQ$y>*f'3'F-7$$!3!)******\\T%e5%F-$\"3v4;&G;DEj'F-7$$!3H+]PM H\\1JF-$\"3-&3%*\\))3(HtF-7$$!3I,++vx*=A#F-$\"3V_(3/$Qj2!)F-7$$!3#z*\\ P%oc8D\"F-$\"30?%*HP=xB))F-7$$!3ax*\\7`?$\"3WNoXI@7y'*F-7$$\"31O +]il6DKp\"F17 $$\"3J(****\\K&**HiF-$\"3q0)fyK7X'=F17$$\"3m)**\\7oLF<(F-$\"3]fU\\d\"R )[?F17$$\"3!)**\\i::)[3)F-$\"3()y0@e?^WAF17$$\"3A)**\\(ohm(4*F-$\"33]3 3-Gu$[#F17$$\"3o****\\A)p2+\"F1$\"3dI=kG_P?FF17$$\"3I++vo^$z4\"F1$\"3K Yv#yKpz*HF17$$\"3y*\\iST\")f=\"F1$\"3)p^T^H)*QF$F17$$\"3E++D;#RAG\"F1$ \"3J>)eLS-Zg$F17$$\"3q*\\ilI5GP\"F1$\"3%3@vAzDk%RF17$$\"3%)*\\7G>$[n9F 1$\"3uz3(zr-$QVF17$$\"3/++vVK/g:F1$\"31(oj)Qq-fZF17$$\"3!)*\\i!R]%pl\" F1$\"3-k&)Hs$oKC&F17$$\"3]+++&)HF]F1$\"3Ur&*)zS$HhpF17$$\"3x****\\K (Rt-#F1$\"3AXZ&4xdQf(F17$$\"3p**\\(oDAq7#F1$\"3%zug(pn%)*Q)F17$$\"3W++ +&\\zh@#F1$\"3I\"3qIK@A<*F17$$\"3m*\\ilqR7J#F1$\"3>HC,t@p35!#;7$$\"3)) *\\P%eWA-CF1$\"3GD36FCx/6F`z7$$\"3++++++++DF1$\"3LZ.2'R\\#=7F`z-%'COLO URG6&%$RGBG$\"*++++\"!\")$F*F*Fb[l-F$6$7S7$F($\"\"%F*7$F/$\"3#****\\i! G\">!RF17$F5$\"34+vVjwc;QF17$F:$\"3')**\\PC\")e?PF17$F?$\"3A+]iqB(Ri$F 17$FD$\"30+vo9e\"y_$F17$FI$\"3++vVyjmQMF17$FN$\"31+v$4IdjM$F17$FS$\"3A +v$47\"*3D$F17$FX$\"33+v=-6tbJF17$Fgn$\"35++DEt%y0$F17$F\\o$\"3/+]7V7j rHF17$Fao$\"3k***\\PNsX(GF17$Ffo$\"3')***\\7$\\6xFF17$F[p$\"3')***\\ik '>$o#F17$F`p$\"31+v$f%)4zf#F17$Fep$\"3)****\\7f&\\'\\#F17$Fjp$\"3w**** *\\T%e5CF17$F_q$\"3\")*\\PMH\\1J#F17$Fdq$\"38++]x(*=AAF17$Fiq$\"3z*\\P %oc8D@F17$F^r$\"3y*\\7`?F17$Fir$\"3'**\\(=P jtZ=F17$F^s$\"3E+]7`6A_\"F17$Fau$\"3=+]7$QL-4\"F17$Ffu$\"3;.++v " 0 "" {MPLTEXT 1 0 141 "a := exp(1);\nb : = -1;\nc := 2;\nDelta := b/ln(a)*a^(c/b);\nevalf(%);\nx1 := -W(0,-1/De lta)/ln(a)-c/b;\n``=evalf(%);\nunassign('Delta','a','b','c'):" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG-%$expG6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"cG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&DeltaG,$*&\"\"\"F'*$)- %$expG6#F'\"\"#F'!\"\"F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+LGN`8!# 5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G,&-%\"WG6#*$)-%$expG6#\"\" \"\"\"#F.!\"\"F/F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"*,W&GW!\" *" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "exp(x)=2-x;\nsolve(%,x);\nevalf(evalf[14](%));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$expG6#%\"xG,&\"\"#\"\"\"F'!\"\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%\"WG6#-%$expG6#\"\"#!\"\"F*\"\"\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+5SaGW!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "History " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 60 "Acknowledgement: This short historica l note comes from ???? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 789 "The history of the Lambert W function goes back to \+ Johann Heinrich Lambert (1728-1777). \nLambert was born the son of a \+ tailor, and was expected by his father to continue in that profession. His early fight for his education is a remarkable story - he had to s ell drawings and writings to his classmates to buy candles for night s tudy, for example - but eventually his talents were recognized and he \+ got a position as tutor in a house which had a decent library. He then was able to educate himself, and went on to produce fundamental disco veries in cartography (the Lambert projection is still in use), hygrom etry, pyrometry, statistics, philosophy (where he is actually more fam ous than as a mathematician), and pure mathematics. He is most noted a s being the first person to prove that " }{XPPEDIT 18 0 "Pi" "6#%#PiG " }{TEXT -1 150 " is irrational, which was an important step in provin g that the classical problem of squaring the circle was impossible by \+ straightedge and compass. \n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 " " {TEXT -1 18 "Code for pictures " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 34 "Code for si gn of derivative table " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 802 "p1 := plot([[[-3.5,-1.25],[2.5,-1. 25]],[[-3.5,.5],[2.5,.5]],\n [[-3.5,1],[2.5,1]],[[-3.5,-1.25],[-3. 5,1]],\n [[-2.5,-1.25],[-2.5,1]],[[-.5,-1.25],[-.5,1]],\n [[.5 ,-1.25],[.5,1]],[[2.5,-1.25],[2.5,1]]],color=black):\np2 := plottools[ arrow]([-2,-.3],[-1,-1],\n 0,.15,.15,arrow,color=black,thicknes s=2):\np3 := plottools[arrow]([1,-1],[2,-.3],\n 0,.15,.15,arrow ,color=black,thickness=2):\nt1 := plots[textplot]([[-3,.75,`x`],[-1.5, .75,`x < -1`],\n [0,.75,`-1`],[1.5,.75,`x > -1`]],color=black,font= [HELVETICA,10]):\nt2 := plots[textplot]([[-3,.2,`g '(x)`],[0,.2,`0`]], \n color=black,font=[HELVETICA,10]):\nt3 := plots[textplot]([[- 1.5,.35,`_`],[1.5,.2,`+`]],\n color=black,font=[HELVETICA,14]): \nplots[display]([p1,p2,p3,t1,t2,t3],axes=none,\n view=[-3.5..2.5 ,-1.3..1.1]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 812 "p1 := plot([[[-3.5,-1.25],[2.5,-1.25]],[[-3.5,. 5],[2.5,.5]],\n [[-3.5,1],[2.5,1]],[[-3.5,-1.25],[-3.5,1]],\n \+ [[-2.5,-1.25],[-2.5,1]],[[-.5,-1.25],[-.5,1]],\n [[.5,-1.25],[.5,1 ]],[[2.5,-1.25],[2.5,1]]],color=black):\np2 := plot([-1.5+.6*cos(t),-1 +.6*sin(t),t=Pi-.2..0.2],\n color=black,thickness=2):\np3 \+ := plot([1.5+.6*cos(t),-.3+.6*sin(t),t=-Pi+.2..-.2],\n col or=black,thickness=2):\nt1 := plots[textplot]([[-3,.75,`x`],[-1.5,.75, `x < -2`],\n [0,.75,`-2`],[1.5,.75,`x > -2`]],color=black,font=[HEL VETICA,10]):\nt2 := plots[textplot]([[-3,.2,`g ''(x)`],[0,.2,`0`]],\n \+ color=black,font=[HELVETICA,10]):\nt3 := plots[textplot]([[-1.5 ,.35,`_`],[1.5,.2,`+`]],\n color=black,font=[HELVETICA,14]):\np lots[display]([p1,p2,p3,t1,t2,t3],axes=none,\n view=[-3.5..2.5,-1 .3..1.1]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 31 "y*exp(y) =x exactly when y=W(x) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 783 "p1 := plot([y*exp(y),y,y=-1..1.23] ,color=red):\np2 := plot([y*exp(y),y,y=-4..-1],color=blue):\np3 := plo t([[[-exp(-1),-1]]$3],style=point,color=[black,green$2],\n symb ol=[circle,diamond,cross]):\nt1 := plots[textplot]([[1.66,1.18,`y = W \+ (x)`]],\n font=[HELVETICA,10],color=COLOR(RGB,.95,0,0)):\nt 2 := plots[textplot]([[1.77,1.1,`0`]],\n font=[HELVETICA,8], color=COLOR(RGB,.95,0,0)):\nt3 := plots[textplot]([[-.8,-2.2,`y = W ( x)`]],\n font=[HELVETICA,10],color=COLOR(RGB,0,0,.95)):\nt4 \+ := plots[textplot]([[-.68,-2.28,`-1`]],\n font=[HELVETICA,8] ,color=COLOR(RGB,0,0,.95)):\nt5 := plots[textplot]([[4.3,-.1,`x`],[-.1 3,1.27,`y`]],\n font=[HELVETICA,10],color=COLOR(RGB,0,0,0)): \nplots[display]([p1,p2,p3,t1,t2,t3,t4,t5],font=[HELVETICA,9]);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 38 "Illusration of solution of exp(x)=x+c " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 522 "x1,x2 := solve(exp(x)=x+1.5):\ny1 := x1+1.5: \+ y2 := x2+1.5:\np1 := plot(exp(x),x=-2..1.7,color=red):\np2 := plot([x+ .5,x+1,x+1.5],x=-2..1.7,color=blue):\np3 := plot([[[0,1],[x1,y1],[x2,y 2]]$3],style=point,color=black,\n symbol=[circle,diamond,cross] ):\nt1 := plots[textplot]([[2.3,2.22,`y = x + 0.5`],\n [2.23,2.73,` y = x + 1`],[2.3,3.2,`y = x + 1.5`]],color=COLOR(RGB,0,0,.9)):\nt2 := \+ plots[textplot]([[2.5,-.15,`x`],[-.15,5.6,`y`]],color=black):\nplots[d isplay]([p1,p2,p3,t1,t2],view=[-2..2.5,-1.6..5.6],labels=[``,``]);" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "6 \+ 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }