Analyzing Complex Function z^a: Derivative & Analytic Region

[Hodgey8806]

Homework Statement

Define z^a = exp(a log z), assume a is complex
Where is this function analytic, and what is its derivative.

Homework Equations

Log z is defined as log z = ln |z| + i*arg z, 0 <= arg z < 2pi.

The Attempt at a Solution

I am really unsure of how to look at this problem.

If I take the derivative, is it incorrect to say that it is 0?

z^a = z^a*ln z*(a' =0) thus, z^a = 0. Is that correct?

If so, then it should be analytic everywhere. Is that correct?

 

[kru_]

The derivative is not 0.

Powers of a complex number are often multiple-valued.

Remember, in order for a function to be differentiable, it must be single valued. So consider the domain where Log is single valued for an arbitrary angle, say rho.

Let $z = re^{i\theta}$, then you have $log(z) = ln|r| + i\theta$ which is singled valued in $(r > 0, \rho < \theta < \rho + 2\pi)$

When you only consider the singlue-valued branch of the log, then you can use rules for derivation to find the derivative.