When does derivative of complex Log exist?

  • Thread starter mnb96
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  • #1
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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]
 

Answers and Replies

  • #2
Cyosis
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Try the Cauchy-Riemann equations.
 
  • #3
713
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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?
 

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