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When does derivative of complex Log exist?

  1. May 30, 2010 #1
    if I have a function [tex]f:\mathbb{R}\rightarrow \mathbb{C}[/tex], what are the conditions on f for which the derivative of the Logarithm exist?

    [tex]\frac{d}{dx}\mathrm{Log} (f(x))[/tex] exists?

    Note that here I defined:

    [tex]Log(z)=\log |z| + Arg(z)[/tex], where [itex]-\pi< Arg(z) \leq \pi[/itex]
  2. jcsd
  3. May 30, 2010 #2


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    Homework Helper

    Try the Cauchy-Riemann equations.
  4. May 30, 2010 #3
    Ok...so, given the fact that Log(z), (with z complex) as defined above, is differentiable in the region:

    [tex]Arg(z)\in (-\pi,\pi)[/tex]

    In order for Log(f(x)) to be differentiable, I need to have that;

    1) the image of f(x) must be in that region (the one where Log is differentiable)
    2) f must satisfy the C-R equations

    Is this correct?
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