- #1

elgen

- 64

- 5

## Homework Statement

This problem is on how to identify the branch points and the branch cuts of a multivalued function. Consider the following function [itex]f(z)=Log(1+z^\alpha)[/itex] where [itex]\alpha[/itex] is a rational number and [itex]z\in \mathcal{C}[/itex].

## Homework Equations

Obviously, for the function [itex]z^\alpha[/itex], it has two branch points at [itex]z=0[/itex] and [itex]z=\infty[/itex]. Also, for the function [itex]Log(z)[/itex], it has two branch points at [itex]z=0[/itex] and [itex]z=\infty[/itex] and the negative real axis is often selected to be the branch cut.

## The Attempt at a Solution

My difficulty lies in identifying the branch points of the cascaded function[itex]f(z)=Log(1+z^\alpha)[/itex]. Let [itex]z=r e^{j\theta}[/itex]. we have

[itex]Log[1+r^\alpha e^{i\alpha(2\pi k+\theta)})=Log(1+r^\alpha\cos(\alpha(2\pi k+\theta))+i r^\alpha\sin(\alpha(2\pi k+\theta))][/itex]. If I let

[itex] 1+r^\alpha\cos(\alpha(2\pi k+\theta)) < 0[/itex] and

[itex]r^\alpha\sin(\alpha(2\pi k+\theta)) =0 [/itex]. I have [itex]\theta=\frac{m\pi}{\alpha}-2\pi k[/itex] and [itex]r^\alpha\cos(m\pi)<-1[/itex].

I get [itex]m[/itex] must odd, and [itex] r>1 [/itex]. This seems to suggest that a branch cut starting at [itex]r>1[/itex] and an angle [itex]\theta=\frac{m\pi}{\alpha}-2\pi k[/itex].

When I plot the function in Mathematica, the branch cut starts at z=0 and extends along the negative real axis. This is in contradiction with my prediction.

How could I identify the branch cut and branch points of this function?

Thank you.Elgen