(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

What is a real holomorphic function which is not analytic?

2. Relevant equations

Theorem from complex analysis: holomorphic functions and analytic functions are the same.

Definition 1: A holomorphic function is infinitely differentiable.

Definition 2: An analytic function is locally given by a convergent power series.

3. The attempt at a solution

I think one answer is [tex]e^{\frac{-1}{x^2}}[/tex]. The function can be differentiated by the chain rule as many times as desired (so it is holomorphic) but has a Taylor series with all coefficients equal to zero (so it is not analytic).

However, I wonder about the complex-valued function [tex]e^{\frac{-1}{z^2}}[/tex]. Does it not have the same problem? That is to say, how can we show that it is consistent with the holomorphic-analytic equivalence theorem if (1) we can differentiate it infinitely many times but (2) it's Taylor series has all coefficients equal to zero?

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# Homework Help: Holomorphic = Analytic

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