Hyperbolic functions

[CaptainBlack]

Bany's question from Yahoo Questions:

Hi guys, I just need some help with a complex variables question
I need to "Let w=f(z)=coth(z/2). Show that w=f(z)=h(C) = (C+1)/(C-1) where C=g(z)=e^z"Thanks guys, any help will be super appreciated and best answer given!

CB

 

[CaptainBlack]

I need to "Let w=f(z)=coth(z/2). Show that w=f(z)=h(C) = (C+1)/(C-1) where C=g(z)=e^z"

Write \(\coth(z/2)\) in exponential form using:

\[\coth(u)=( e^u + e^{-u} )/( e^u - e^{-u} )\]

Then:

\[f(z)=( e^{z/2} + e^{-z/2} )/( e^{z/2} - e^{-z/2} )\]

now multiply top and bottom by \(e^{z/2}\) to get:

\[f(z)=( e^{z} + 1 )/( e^{z} - 1 )\]

so when you substitute \(C=e^{z}\) you are done.

CB