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{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(`ev
alf/invGAMMA expects its argument to be                       not smal
ler 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#:6#%\"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(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 385 "`mysi
mplify/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,functio
n) and op(0,g)=GAMMA then\n      h:=op(1,g);\n      if type(h,function
) 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;" "/*$-%%sqrtG6#%\"xG\"\"#F'" }
}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "sqrt(x^2) = x;" "/-%%sqrtG6#*$
%\"xG\"\"#F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 421 "`mysimplif
y/GAMMA/symbolic`:=proc(f)\n  local g,h;\n  if type(f,\{table, procedu
re, 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,func
tion) then\n      h:=op(1,g);\n      if type(h,function) 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 "`mysim
plify/GAMMA/symbolic`(GAMMA(invGAMMA(x)));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "`my
simplify/GAMMA/symbolic`(invGAMMA(GAMMA(x)));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "`my
simplify/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(invGAMMA(a))*di
ff(a,x)\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "diff(invGA
MMA(t*exp(t)),t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**%\"tG!\"\"-%$ex
pG6#F$F%-%$PsiG6#-%)invGAMMAG6#*&F$\"\"\"F&F0F%,&F&F0F/F0F0" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "asympt(ln(GAMMA(z)),z,5);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&,&-%#lnG6#%\"zG\"\"\"!\"\"F*F*F)F
*F*-F'6#*&\"\"##F*F/%#PiGF0F*F&#F+F/*$F)F+#F*\"#7-%\"OG6#*$F)!\"$F*" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "lb:=convert(asympt(ln(GAMM
A(z)),z,3),polynom);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#lbG,(*&,&-%
#lnG6#%\"zG\"\"\"!\"\"F,F,F+F,F,-F)6#*&\"\"##F,F1%#PiGF2F,F(#F-F1" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "ub:=convert(asympt(ln(GAMMA(
z)),z,5),polynom);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ubG,**&,&-%#l
nG6#%\"zG\"\"\"!\"\"F,F,F+F,F,-F)6#*&\"\"##F,F1%#PiGF2F,F(#F-F1*$F+F-#
F,\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "triple:=[lb,ln(GA
MMA(z)),ub];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'tripleG7%,(*&,&-%#l
nG6#%\"zG\"\"\"!\"\"F-F-F,F-F--F*6#*&\"\"##F-F2%#PiGF3F-F)#F.F2-F*6#-%
&GAMMAGF+,*F'F-F/F-F)F5*$F,F.#F-\"#7" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 61 "plot(triple,z=2..10,color=[blue,red,blue],thickness=[
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0 0 0 0 0 1 1 0 0 0 0 -6984 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 18 0 "ln(GAMMA(z)) = (z-1/2)*ln(z)-z+ln(2*Pi)/2+2*Int(arctan(
t/z)/(exp(2*Pi*t)-1),t = 0 .. infinity);" "/-%#lnG6#-%&GAMMAG6#%\"zG,*
*&,&F)\"\"\"*&\"\"\"F-\"\"#!\"\"F1F--F$6#F)F-F-F)F1*&-F$6#*&\"\"#F-%#P
iGF-F-\"\"#F1F-*&\"\"#F--%$IntG6$*&-%'arctanG6#*&%\"tGF-F)F1F-,&-%$exp
G6#*(\"\"#F-F9F-FEF-F-\"\"\"F1F1/FE;\"\"!%)infinityGF-F-" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "2*int(t/(exp(2*Pi*t)-1),t=0..infini
ty);" }}{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;" "/*&%\"zG\"\"\"-%$expG6#F$F%%\"xG" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "z = ln(x)-ln(z);" "/%\"zG,&-%#ln
G6#%\"xG\"\"\"-F&6#F#!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "z
 = ln(x)-ln(ln(x)-ln(z));" "/%\"zG,&-%#lnG6#%\"xG\"\"\"-F&6#,&-F&6#F(F
)-F&6#F#!\"\"F1" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "z = ln(x)-ln
(ln(x))-ln(1-ln(z)/ln(x));" "/%\"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)F1
F1F(F$!\"$#F(F5*(F)F(,**$F)\"\"$F<F6!#AF)\"#O!#7F(F(F$!\"%#F(\"#7*(F)F
(,,F;!$D\"F6\"$]$*$F)\"\"%FBF)!$+$\"#gF(F(F$!\"&#F(FJ-%\"OG6#*$F$!\"'F
(" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "GAMMA(z) = x;" "/-%&GAMMAG
6#%\"zG%\"xG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "(z/exp(1))^z*sq
rt(2*Pi/z)(1+etc) = x;" "/*&)*&%\"zG\"\"\"-%$expG6#\"\"\"!\"\"F&F'--%%
sqrtG6#*(\"\"#F'%#PiGF'F&F,6#,&\"\"\"F'%$etcGF'F'%\"xG" }}}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 18 0 "z*ln(z)-z-ln(z)/2+ln(sqrt(2*Pi))+etc = \+
L[1];" "/,,*&%\"zG\"\"\"-%#lnG6#F%F&F&F%!\"\"*&-F(6#F%F&\"\"#F*F*-F(6#
-%%sqrtG6#*&\"\"#F&%#PiGF&F&%$etcGF&&%\"LG6#\"\"\"" }}}{EXCHG {PARA 0 
"" 0 "" {XPPEDIT 18 0 "ln(z)+ln(ln(z))+ln(1-1/ln(z)-1/(2*z)+etc) = L[2
];" "/,(-%#lnG6#%\"zG\"\"\"-F%6#-F%6#F'F(-F%6#,*\"\"\"F(*&\"\"\"F(-F%6
#F'!\"\"F5*&\"\"\"F(*&\"\"#F(F'F(F5F5%$etcGF(F(&%\"LG6#\"\"#" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "ln(z) = L[2]-ln(ln(z))-ln(1-1/ln
(z)-1/(2*z)+etc);" "/-%#lnG6#%\"zG,(&%\"LG6#\"\"#\"\"\"-F$6#-F$6#F&!\"
\"-F$6#,*\"\"\"F,*&\"\"\"F,-F$6#F&F1F1*&\"\"\"F,*&\"\"#F,F&F,F1F1%$etc
GF,F1" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "lambda = ln(z);" "/%'l
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{XPPEDIT 18 0 "lambda = L[2]-ln(lambda)-ln(1-1/lambda);" "/%'lambdaG,(
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}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "lambda = L[2]-L[3]+Sum(P[k](L[
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