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Determining an Analytic Function from its Real Part

  1. Apr 22, 2012 #1
    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.

    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.
     
  2. jcsd
  3. Apr 23, 2012 #2
    In case of an analytic function, the real part and the imaginary part are related thanks to the Kramers-Kroning relationships.
    So, if you know the real part function, you can compute the imaginary part function. Consequently, the complex function is obtained.
    http://en.wikipedia.org/wiki/Kramers–Kronig_relations
     
  4. Apr 23, 2012 #3

    Bacle2

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    Poopsilon, you may want to be careful with your layout; the real part of logz is ln|z|, not

    log|z|. Then you would have log|z|=ln||z||+iarg|z|=ln|z|+iarg|z|. But |z| is real-valued.

    So, in a sense, ln|z| is a function of a real argument, and, unlike logz, it is radially-

    constant; same for arg|z|. Have you double-checked that log|z| is actually harmonic, and

    what its domain of harmonicity is? Maybe Wolfram has a way of helping you double-check.
     
  5. Apr 23, 2012 #4

    Bacle2

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    I just realized I may have misunderstood (misunderestimated?) your OP. When you

    write log|z|: is this the real log ,usually written ln|z|, or is it the complex log?

    For one thing, log|z| as a real-valued function is not even defined for z=0.
     
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