- #1

Raziel2701

- 128

- 0

## Homework Statement

Define a single-valued branch of the function [tex]f(z) =z^z[/tex] on an open set [tex]U\subseteq C[/tex], show f is analytic on U, and find f'(z)

## The Attempt at a Solution

[tex] z^z = e^{zlog(z)}[/tex] So because of the log I have to define or pick a branch where f(z) is defined. Since f(z) is more or less an increasing exponential function that starts from the y axis, could I just pick x>0 and y>0 for my branch?

I tried substituting x+iy for z, so that I may break the function apart into its real and imaginary parts to see if they satisfy the Cauchy-Riemann equations to test f(z) if it's analytic, but then I get a mess:

[tex]e^{(x+iy)(log(\sqrt{x^2 + y^2}) +i arg(x+iy)}[/tex]

And I didn't pursue it because I don't even know if this is the right approach.

Also, to find f'(z), I get z^z(log(z) +1)). Is this right?