Billy 's question at Yahoo Answers (Complex analysis)

[Fernando Revilla]

Here is the question:

write the function f(z)=z+1/z (z cannot = 0)
in the form f(z)=u(r,theta) + iv(r,theta)

Here is a link to the question:

Complex analysis help? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.

 

[Fernando Revilla]

Hello Billy,

Using $z=r(\cos\theta+i\sin\theta)=re^{i\theta}$ and if $z\neq 0$ (i.e. $r\neq 0)$:
$$\begin{aligned}f(z)&=z+\frac{1}{z}\\&=re^{ i\theta}+\frac{1}{re^{i\theta}}\\&=re^{i\theta}+ \frac{1}{r}e^{-i\theta}\\&=r(\cos \theta+i\sin\theta)+ \frac{1}{r}(\cos \theta -i\sin\theta)\\&= \left (r+\frac{1}{r}\right)\cos\theta +i\left (r-\frac{1}{r}\right)\sin\theta\end{aligned}$$