In article <324FEC1A.email@example.com>, David Ullrich <firstname.lastname@example.org> wrote: >Christopher wrote: >> >> Hello, >> Please tell me how to prove simply that an analytic function is a function of >> z only, no z_bar. > > Using z* for the conjugate of z: > > You need to state the conclusion more precisely before this can be >proved. We all agree that f(z) = z^2 is analytic, but f(z) = g(z*) for a >certain function g , so f _is_ "a function of z*".
Yes, of course this is right.
For any function of z, say, y = f(z) = f(conj(z*)).
So, how about the following way to say this.
If y = f(z) is an analytic function of z then y is necessarily not an analytic function of z*, and if y = g(z*) is an analytic function of z*, then y is necessarily not a function of z.
In either case, I think you could use chain rule would show that dz*/dz exists, which can be easily shown to be false.
If y is an analytic function of z then dy/dz = f'(z)