I've posted before about my fractal project, and this is something else that is related. The function:

where

.

The derivative along this line:

for a new real variable

.

I'm trying to find when the real and/or imaginary parts equal zero.

Is there any other way to solve these equations without splitting

into

and

just because it is so messy and long? I know one method of removing the imaginary terms from the bottom by multiplying top and bottom by

, factoring

out of the denominator, and multiplying numerator and denominator by

resulting in a denominator of

. It's still messy with a numerator of

but now, algebraically, the worst part is

. I'd appreciate any suggestions.