Why are z^(1/3), z^(1/2) and ln(z) not unique in the complex plane?

[DanielO_o]

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]

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 $e^{i\theta/3}$.

 

[DanielO_o]

Thanks :)

 

[bobmerhebi]

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 please tell me.

Thank You in Advance

----------------
Yours Truly
BOB Merhebi
Astrobob Group
www.astrobob.tk[/URL]