# Thread: simplifying nasty complex math expression

1. ## simplifying nasty complex math expression

$\frac{4+3j}{200}(\frac{-3+4j}{25})^n+\frac{4-3j}{200}(\frac{-3-4j}{25})^n$

I know 100% that this expression must be real since the excitation to the system was real in my Linear, time-invariant system of differential equations.

Somehow, the solutions manual transforms the equation into some e^(jw)s, and uses Euler's identity from there to reach two real sinusoidal functions. The problem I'm having is making the e^(jw) out of that expression in the first place.

2. Ok, you have some exponentiation going on there, twice. What I would do is convert the rectangular representations you have into polar ones via the following: if you have

$z=x+jy$, then

$z=r\,e^{j\theta}$, where

$r=\sqrt{x^{2}+y^{2}}$, and

$\theta=\tan^{-1}\!\left(\frac{y}{x}\right)$.

Exponentiation, and multiplication for that matter, is easier in the polar representation of complex numbers than in the rectangular. However, I would definitely use rectangular for addition and subtraction. So, I would get those numbers into polar, do the exponentiation, convert back to rectangular, and see what you get.

3. ahhh, of course. Thanks.

4. No problem. Have fun!