# Complex Analysis

How does one show that z^{1/3} is not unique in the complex plane?

[ Similarly for z^(1/2) and ln(Z) ]

Thanks,

Daniel

HallsofIvy
Homework Helper
Write z as $re^{i\theta}$ in "polar form". Then $z^{1/3}= r^{1/3}e^{i\theta/3}$. Now $e^{i(\theta+ 2\pi)}= e^{i\theta}$ but $e^{i(\theta+ 2\pi)/3}$ is not the same as $e^{i\theta/3}$.

Thanks :)

Dear Mentors,

Could anyone include explanations about the Laurent series, & the Residues & Poles ? Everything for an undergraduate course ?

I didn't find anything about that on the forum. if there's a good one plz tell me.