Im making use of many different formulas expressed using Complex variables
(A well known example would be the mandelbrot fractal.. z_n+1 = z_n^2 + c)
And I want to find a source that describes every known algebraic, trigonometric, or other form of mathematical operation that has a corresponding form in the Complex plane that differs in behavior to real numbers.
For example... a * b as real numbers can be expressed as (a.r * b.r - a.i * b.i) + (a.r * b.i + a.i * b.r)i if you do the operations on the real and imaginary parts in a single formula.
Or rather, if you were to use complex numbers as vectors / arrays, which is common in a computer environment.. it would likely be expressed like this:
vec2(a.x * b.x - a.y * b.y, a.x * b.y + a.y * b.x)
Using those conversions, you can express any complex formula as an algebraic operation on real number 2D vectors. (Such as the above aforementioned mandelbrot fractal, which, iterated repeatedly, would be represented as z = vec2(z.x * z.x - z.y * z.y, 2.0 * z.x * z.y) + c in a programming language)
My question is, where can I find a source for these CAS formulas for all known complex operations.. And while it might not be totally on topic for this site, I was wondering if there was a well known tool or application that can automatically convert Complex formulas into CAS Formulas I can then run locally on any programming language.