When does derivative of complex Log exist?

In summary, the conditions for the derivative of the Logarithm to exist for a function f:\mathbb{R}\rightarrow \mathbb{C} are that the image of f(x) must be in the region where Log is differentiable (|z|>0, Arg(z)\in (-\pi,\pi)), and f must satisfy the Cauchy-Riemann equations.
  • #1
mnb96
715
5
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]
 
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  • #2
Try the Cauchy-Riemann equations.
 
  • #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?
 

1. What is a derivative of complex Log?

A derivative of complex Log refers to the rate of change of a complex Log function, which is a logarithmic function involving complex numbers. It measures how the output of the function changes with respect to changes in the input.

2. When does the derivative of complex Log exist?

The derivative of complex Log exists when the complex Log function is differentiable at a given point. This means that the function is continuous and has a defined tangent line at that point.

3. How is the derivative of complex Log calculated?

The derivative of complex Log can be calculated using the complex differentiation rules, which involve taking the partial derivatives of the real and imaginary parts of the complex Log function.

4. Can the derivative of complex Log be undefined?

Yes, the derivative of complex Log can be undefined at certain points, such as when the complex Log function is not differentiable. This can occur at branch points or singularities where the function is discontinuous or has a vertical tangent.

5. Why is the derivative of complex Log important?

The derivative of complex Log is important in many areas of mathematics, including complex analysis, differential geometry, and physics. It allows us to analyze the behavior of complex Log functions and solve problems related to optimization, curve sketching, and differential equations.

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