Help with Hyperbolic Functions

CaptainBlack

Well-known member
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

Well-known member
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