(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

##z## is a complex number such that ##z = \frac{a+bi}{a-bi}##, where ##a## and ##b## are real numbers. Prove that ##\frac{z^2+1}{2z} = \frac{a^2-b^2}{a^2+b^2}##.

2. Relevant equations

3. The attempt at a solution

I calculated

\begin{equation*}

\begin{split}

z = \frac{a+bi}{a-bi} &= \frac{a+bi}{a-bi}\times \frac{a+bi}{a+bi} \\

&= \frac{a^2+2abi-b^2}{a^2+b^2} \\

&= \frac{a^2-b^2}{a^2+b^2}+\frac{2ab}{a^2+b^2}i.

\end{split}

\end{equation*}

But sticking that ugly thing into ##\frac{z^2+1}{2z}## gives me something nasty. I'm sure there is a much simpler way!!!

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Complex Proof

Have something to add?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**