Pibe wrote:

I’m sorry, but my english is very bad.

Actually what you wrote is pretty good. I haven't checked your proof yet - that might take a bit longer, and someone else might check it before me.

However here's a version with the English corrected:

I’m an engineer but I’m working as a teacher at a high school in Spain.

I’m not sure about my proof, but here it is.

Let

with z=const. Then,

and I'll write the Maclaurin series for g(p):

Now, let z=variable, p=const, and let's multiply everything by f(z):

and then (without rigorous justification),

But if we use the Cauchy formula in the 'origin shifted' form, that is

, then

And the integral

is the 'definition' of the k-th derivative at the origin, so

Now, we take p=variable and rename it as 'z':

So, f(z) is analytic Q.E.D.

Please, could you correct my grammatical errors and tell me if this proof is right?!

Thank you.