Writing a function in u+iv form

1. Jul 13, 2012

unchained1978

This is related to another post of mine. How would you go about writing $\frac{1}{e^{z}-1}$ in u+iv form? Usually multiplying through with the complex conjugate gives you the desired form, but here I'm not sure that it works. Any suggestions?

2. Jul 13, 2012

haruspex

Write z = x+iy. Massage the denominator into u+iv form, then multiply top and bottom by conjugate.

3. Jul 13, 2012

DonAntonio

With $\,z=x+iy\,\,,\,x,y,\in \Bbb R\,\,\,and\,\,\,e^{ix}=\cos x+i\sin x$ :

$$\frac{1}{e^z-1}=\frac{1}{e^x\cos y -1+ie^x\sin y}=\frac{e^x\cos y-1-e^xi\sin y}{e^{2x}-2e^x\cos y+1}=\frac{e^x\cos y-1}{e^{2x}-2e^x\cos y+1}-\frac{e^x\sin y}{e^{2x}-2e^x\cos y+1}\,i$$

Piece of cake (pant,pant!)

DonAntonio

4. Jul 13, 2012

Staff: Mentor

Write ez as ex + iy = exeiy = ex(cosy + i siny). Your denominator is this expression, minus 1.

It's slightly messy, but you can rationalize the denominator by multiplying by the conjugate over itself.