I'm trying to prove that log|z| is not the real part of an analytic function defined on an annulus centered at zero. Due to the Cauchy-Riemann Equations, I've been under the impression that given a harmonic function, such as log|z|, its role as the real part of an analytic function is unique, and thus an analytic function is completely determined, up to the addition of a constant, by its real (or imaginary) part.(adsbygoogle = window.adsbygoogle || []).push({});

Thus I feel like since log(z) is an analytic function which cannot be defined on an annulus centered at zero whose real part is log|z|, then I can conclude that log|z| is not the real part of an analytic function defined on said annulus.

Can someone help clarify my understanding? Thanks.

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# Determining an Analytic Function from its Real Part

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