#Math640 (Rutgers), Spring 2016, Feb. 29, 2016
#C12.txt, starting the Jacobian conjecture
Help:=proc(): print(` Inv(f,z,N) , LIF(f,z,N) `): end:

#Inv(f,z,N): inputs an expression f in z (denoting a function of z that
#vanishes at z=0 f(0)=0, tries to find the first N terms of
#the inverse function (formal power series) g(z) such that
#f(g(z))=z and g(f(z))=z
Inv:=proc(f,z,N) local f1,g,a,gN:
option remember:

if eval(subs(z=0,f))<>0 then
  RETURN(FAIL):
fi:

a:=eval(subs(z=0,diff(f,z))):

if a=0 then
 RETURN(FAIL):
fi:



if N=1 then
 RETURN(z/a):
fi:


g:=Inv(f,z,N-1)+gN*z^N:

eval(subs(solve(coeff(taylor(subs(z=f,g),z=0,N+1),z,N),{gN}),g)):

end:

#LIF(f,z,N): does the same as Inv(f,z,N), but much slower, using the Lagrange Inversion Formula
LIF:=proc(f,z,N) local g,n,c1:


c1:=subs(z=0,normal(z/f)):

g:=c1*z:

for n from 2 to N do
 c1:= subs(z=0,normal(diff((z/f)^n,z$(n-1))))/n!:
 g:=g+c1*z^n:
od:

g:

end: