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.
To Dan, thanks. I have gotten thus far before in other equations and have used a formula to separate into x and y, but it is so large and not worth it.
To RonL, I should have specified, I'm looking for the intercept of the real axis and the intercept of the imaginary axis. That point is the cusp of the curve . Thanks anyway.
I stumbled upon the identity
and because is periodic
leading to and thus I need to solve , which hopefully won't call for me to find x and y.
The other equation can also be done in the same manor. I'll tell you how this goes and thanks again.
Oh and a note, I substituted . I should have said that in the original post.