Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

When does derivative of complex Log exist?

  1. May 30, 2010 #1
    Hello,
    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

    Cyosis

    User Avatar
    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]|z|>0[/tex]
    [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?
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook