{VERSION 3 0 "DEC ALPHA UNIX" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{PSTYLE " Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 568 "invGAMMA:=proc(x::a lgebraic)\n local z,f,n;\n if type(x,`!`) then op(1,x)+1\n elif typ e(x,posint) then\n z:=`evalf/invGAMMA`(x);\n f:=1;\n for n from 2 to z do\n f:=n*f;\n if f=x then RETURN(n+1) fi\n od;\n \+ 'invGAMMA(x)'\n elif type(x/Pi^(1/2),rational) and x/Pi^(1/2)>1 then \n z:=`evalf/invGAMMA`(x);\n f:=1/2;\n for n from 2 to z do\n f:=(n-1/2)*f;\n if f=x/Pi^(1/2) then RETURN(n+1/2) fi\n od; \n 'invGAMMA(x)'\n elif type(x,float) then\n `evalf/invGAMMA`(x) \n else\n 'invGAMMA'(x)\n fi\nend:\ninvGAMMA(1):=2:\ninvGAMMA(3/4 *Pi^(1/2)):=5/2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 251 "`evalf /invGAMMA`:=proc(x::algebraic)\n local xx;\n xx:=evalf(x);\n if xx> =1 then \n fsolve(GAMMA(t)=xx,t=2..infinity)\n else\n ERROR(\"e valf/invGAMMA expects its argument to be not sma ller than 1, but received \",x)\n fi\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "seq(invGAMMA(n),n=1..6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(\"\"#\"\"$-%)invGAMMAG6#F$-F&6#\"\"%-F&6#\"\"&F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "invGAMMA(N!);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,&%\"NG\"\"\"F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "seq(invGAMMA(GAMMA(n+1/2)),n=2..5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&#\"\"&\"\"##\"\"(F%#\"\"*F%#\"#6F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "evalf(invGAMMA(10));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+^w2!R%!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "invGAMMA(10.);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+^w2!R%!\"* " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "GAMMA(invGAMMA(x));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%&GAMMAG6#-%)invGAMMAG6#%\"xG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "subsop(3=NULL,readlib(`simpl ify/GAMMA`));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#R6#%\"fG6#%\"gG6\"F(C )>8$--%(readlibG6#.%-normal/GAMMAG6#9$>F+-%(collectG6%F+%&GAMMAG%'norm alG>F+-%7simplify/GAMMA/combineG6#F+>F+F5>F+-%8simplify/GAMMA/products GF=>F+-F-F=F5F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 385 "`my simplify/GAMMA`:=proc(f)\n local g,h;\n if type(f,\{table, procedure , name, numeric\}) then f\n elif not has(f,\{GAMMA,invGAMMA\}) then f \n else \n g:=readlib(`simplify/GAMMA`)(f);\n if type(g,functi on) and op(0,g)=GAMMA then\n h:=op(1,g);\n if type(h,functio n) and op(0,h)=invGAMMA then\n op(1,h)\n else g\n fi \n else\n map(procname,g)\n fi\n fi\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "`mysimplify/GAMMA`(GAMMA(invGAMMA(x )));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "invGAMMA(GAMMA(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%)invGAMMAG6#-%&GAMMAG6#%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "`mysimplify/GAMMA`(invGAMMA(GAMMA(x)));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%)invGAMMAG6#-%&GAMMAG6#%\"xG" }}} {EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "sqrt(x)^2 = x;" "6#/*$-%%sqrtG6# %\"xG\"\"#F(" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "sqrt(x^2) = x; " "6#/-%%sqrtG6#*$%\"xG\"\"#F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 421 "`mysimplify/GAMMA/symbolic`:=proc(f)\n local g,h;\n if type (f,\{table, procedure, name, numeric\}) then f\n elif not has(f,\{GAM MA,invGAMMA\}) then f \n else \n g:=readlib(`simplify/GAMMA`)(f); \n if type(g,function) then\n h:=op(1,g);\n if type(h,fun ction) then\n if \{op(0,g),op(0,h)\}=\{GAMMA,invGAMMA\} then\n \+ op(1,h)\n else g\n fi\n fi\n else\n \+ map(procname,g)\n fi\n fi\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "`mysimplify/GAMMA/symbolic`(GAMMA(invGAMMA(x)));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "`mysimplify/GAMMA/symbolic`(invGAMMA(GAMMA(x)));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "`mysimplify/GAMMA/symbolic`(invGAMMA(x*GAMMA(x)));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"xG\"\"\"F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "`diff/invGAMMA`:=proc(a,x)\n 1/a/Psi(invGA MMA(a))*diff(a,x)\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 " diff(invGAMMA(t*exp(t)),t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&-%$ expG6#%\"tG\"\"\"*&F(F)F%F)F)\"\"\"*(F(\"\"\"F%\"\"\"-%$PsiG6#-%)invGA MMAG6#F*\"\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "asym pt(ln(GAMMA(z)),z,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&,&-%#lnG6 #%\"zG\"\"\"!\"\"F*F*F)F*F*-F'6#*&-%%sqrtG6#\"\"#\"\"\"-F06#%#PiGF3F*F &#F+F2*&F3F3F)!\"\"#F*\"#7-%\"OG6#*&F3F3*$)F)\"\"$F3F9F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "lb:=convert(asympt(ln(GAMMA(z)),z,3 ),polynom);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#lbG,(*&,&-%#lnG6#%\" zG\"\"\"!\"\"F,F,F+F,F,-F)6#*&-%%sqrtG6#\"\"#\"\"\"-F26#%#PiGF5F,F(#F- F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "ub:=convert(asympt(ln (GAMMA(z)),z,5),polynom);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ubG,** &,&-%#lnG6#%\"zG\"\"\"!\"\"F,F,F+F,F,-F)6#*&-%%sqrtG6#\"\"#\"\"\"-F26# %#PiGF5F,F(#F-F4*&F5F5F+!\"\"#F,\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "triple:=[lb,ln(GAMMA(z)),ub];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'tripleG7%,(*&,&-%#lnG6#%\"zG\"\"\"!\"\"F-F-F,F-F--F* 6#*&-%%sqrtG6#\"\"#\"\"\"-F36#%#PiGF6F-F)#F.F5-F*6#-%&GAMMAGF+,*F'F-F/ F-F)F:*&F6F6F,!\"\"#F-\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 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\"3$Rn=!*yI,-\"F17$FY$\"35^-N)Q\"*>?\"F17$Fhn$\"3;M[OW)=*)R\"F17$F]o$ \"3@!Gt8uH-e\"F17$Fbo$\"3Ge\\EFXr#z\"F17$Fgo$\"3J`e-$>vX,#F17$F\\p$\"3 $f&\\!HCqgB#F17$Fap$\"3\\v#3ZGsMW#F17$Ffp$\"3F`5k([Lvp#F17$F[q$\"39V! \\dLP)=HF17$F`q$\"3\"G7\"\\t.,$=$F17$Feq$\"3xS$znr#oAMF17$Fjq$\"3U xr\"p$F17$F_r$\"3'o8voK7O&RF17$Fdr$\"3i\"eWli4EB%F17$Fir$\"3<:iUmS\"Q \\%F17$F^s$\"3k7FZ\"=c2y%F17$Fcs$\"3j6h3FoM%3&F17$Fhs$\"3[B+TaN3``F17$ F]t$\"3EM3j5!eyk&F17$Fbt$\"3\")p]\"zj'>dfF17$Fgt$\"3#y9gxvBWE'F17$F\\u $\"3$HV!3$4)*ec'F17$Fau$\"3R9s^'ol`!pF17$Ffu$\"3vGUC$yKX@(F17$F[v$\"3> !fXDk_)[vF17$F`v$\"3flF$)=0ZbyF17$Fev$\"3^YIz>&*e%>)F17$Fjv$\"3U?eP/fJ <&)F17$F_w$\"3^n@-)p[$e))F17$Fdw$\"3.3>Rk&R`>*F17$Fiw$\"3\"*p?\"[cP=b* F17$F^x$\"3M+FvE0p)*)*F17$Fcx$\"3l!3+7\"zoD5Fgx7$Fix$\"3)=H;P&3bh5Fgx7 $F^y$\"3+he)\\^)z%4\"Fgx7$Fcy$\"3K]M#G(HCL6Fgx7$Fhy$\"383CnQ\"Hz;\"Fgx 7$F]z$\"3omJ(4/>_?\"Fgx7$Fbz$\"3B())z3#>?T7Fgx7$Fgz$\"34Z\"3![F=!G\"Fg x-F\\[l6&F^[lF_[lF*F*-Fc[l6#\"\"$-F$6%7S7$F($\"3dUPd7rqfK!#@7$F/$\"3b. #)H/jgW$)F-7$F5$\"3/y3([$)fCq\"F97$F;$\"3:SmQO%)eDGF97$F@$\"3E;')RRf!3 5%F97$FE$\"3.4cf+T,.bF97$FJ$\"3Yr9D^jM8pF97$FO$\"3x0^D3;Jy%)F97$FT$\"3 OEz`zS??5F17$FY$\"3F/G9*pa??\"F17$Fhn$\"3J*\\(3*ot*)R\"F17$F]o$\"3(*oL FL$y-e\"F17$Fbo$\"3PE%[W;dFz\"F17$Fgo$\"39LU5*y7Y,#F17$F\\p$\"3eW#[Wq. hB#F17$Fap$\"3JS`4+D]VCF17$Ffp$\"3ylv8m.c(p#F17$F[q$\"3:Ns(4wh)=HF17$F `q$\"3\"*z#efIKI=$F17$Feq$\"31h`tA*)GO&RF17$Fdr$\"3H/DBUZiKUF17$Fir$\"3!Qv9g*z#Q\\%F17$F^s$\"3?U _jm*o2y%F17$Fcs$\"3qXFL_&eV3&F17$Fhs$\"3:-cD]W4``F17$F]t$\"3QGK='4oyk& F17$Fbt$\"3#\\f&\\of?dfF17$Fgt$\"3BRJ)3UKWE'F17$F\\u$\"3o0xgph!fc'F17$ Fau$\"3>!R.@)F17$Fjv$\"3Ha+l88K<&)F17$F_w$\"3l(zA 0y`$e))F17$Fdw$\"3B=kN_VM&>*F17$Fiw$\"3'GO&Ho?%=b*F17$F^x$\"3Wl,:xZp)* )*F17$Fcx$\"38B3G7$)oD5Fgx7$Fix$\"3#4W\"yK7bh5Fgx7$F^y$\"3*=g=_())z%4 \"Fgx7$Fcy$\"39/X(HJVK8\"Fgx7$Fhy$\"3x**p5i%Hz;\"Fgx7$F]z$\"3%p()oxM>_ ?\"Fgx7$Fbz$\"3@T " 0 "" {MPLTEXT 1 0 39 "2*int(t/(exp(2*Pi*t)-1),t=0..infinity);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "asympt(LambertW(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%)LambertWG6#%\"xG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "z*exp(z) = x;" "6#/*&%\"zG\"\"\"-%$expG6#F%F&%\"xG" }}} {EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "z = ln(x)-ln(z);" "6#/%\"zG,&-%# lnG6#%\"xG\"\"\"-F'6#F$!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "z = ln(x)-ln(ln(x)-ln(z));" "6#/%\"zG,&-%#lnG6#%\"xG\"\"\"-F'6#,&-F'6 #F)F*-F'6#F$!\"\"F2" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "z = ln(x )-ln(ln(x))-ln(1-ln(z)/ln(x));" "6#/%\"zG,(-%#lnG6#%\"xG\"\"\"-F'6#-F' 6#F)!\"\"-F'6#,&\"\"\"F**&-F'6#F$F*-F'6#F)F/F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "subs(L[1]=ln(x),asympt(LambertW(exp(L[1])),L[ 1]));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,2-%#lnG6#%\"xG\"\"\"-F%6#F$! \"\"*&F)\"\"\"F$!\"\"F(*&*&F)F(,&F)F(!\"#F(F(F-*$)F$\"\"#F-F.#F(\"\"#* &*&F)F-,(F)!\"*\"\"'F(*$)F)F7F-F7F(F-*$)F$\"\"$F-F.#F(F<*&*&F)F-,**$)F )\"\"$F-FHF=!#AF)\"#O!#7F(F(F-*$)F$\"\"%F-F.#F(\"#7*&*&F)F-,,FF!$D\"F= \"$]$*$)F)\"\"%F-FPF)!$+$\"#gF(F(F-*$)F$\"\"&F-F.#F(FZ-%\"OG6#*&F-F-*$ )F$\"\"'F-F.F(" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "GAMMA(z) = x; " "6#/-%&GAMMAG6#%\"zG%\"xG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 " (z/exp(1))^z*sqrt(2*Pi/z)(1+etc) = x;" "6#/*&)*&%\"zG\"\"\"-%$expG6#\" \"\"!\"\"F'F(--%%sqrtG6#*(\"\"#F(%#PiGF(F'F-6#,&\"\"\"F(%$etcGF(F(%\"x G" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "z*ln(z)-z-ln(z)/2+ln(sqrt( 2*Pi))+etc = L[1];" "6#/,,*&%\"zG\"\"\"-%#lnG6#F&F'F'F&!\"\"*&-F)6#F&F '\"\"#F+F+-F)6#-%%sqrtG6#*&\"\"#F'%#PiGF'F'%$etcGF'&%\"LG6#\"\"\"" }}} {EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "ln(z)+ln(ln(z))+ln(1-1/ln(z)-1/( 2*z)+etc) = L[2];" "6#/,(-%#lnG6#%\"zG\"\"\"-F&6#-F&6#F(F)-F&6#,*\"\" \"F)*&\"\"\"F)-F&6#F(!\"\"F6*&\"\"\"F)*&\"\"#F)F(F)F6F6%$etcGF)F)&%\"L G6#\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "ln(z) = L[2]-ln(ln( z))-ln(1-1/ln(z)-1/(2*z)+etc);" "6#/-%#lnG6#%\"zG,(&%\"LG6#\"\"#\"\"\" -F%6#-F%6#F'!\"\"-F%6#,*\"\"\"F-*&\"\"\"F--F%6#F'F2F2*&\"\"\"F-*&\"\"# 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