# Derivating polynomial with complex argument

1. Mar 28, 2007

### Alesak

hello all(where is some waving smiley?)

ive got one problem. i have a polynomial, i.e. Ax^n, but x is a complex number. so its A(a+bi)^n and a and b are variables. then i need partial derivative of a with respect to b. so i can simply rewrite it using binomial theorem to A(a^n*b^0*i^0 + ... + a^0*b^n*i^n) and threat with b as with constant and derivate it as ordinary polynomial. its easy to do it in hand but i need to do it in a program in which its really difficult to do. i was thinking to use some other form of complex number, as A|x|(e^i*(angle)*n) or A|x|(cos(angle*n)+i*sin(angle*n)), which is easy to exponentiate but hard to derivate(algebraic form is hard to exponentiate and easy to derivate, how sweet). im asking if i can derivate complex polynomial somehow else than as i said or substituing for |x| (a*a+b*b)^0.5 and for angle inv. tan(b/a) using some polar form of complex number.

i hope someone understood my confused explanation with my crappy english. im even not sure if all my thoughts are correct. maybe tommorow in school i´ll figure something out. and its no homework, im doing it just for fun.

2. Mar 28, 2007

### HallsofIvy

Staff Emeritus
You could also use the chain rule. If I understand this correctly, you have a polynomial, P(z), of a complex variable, z= a+ ib, and you want to find the derivative of P with respect to b. That is dP/db= (dP/dz)(dz/db)= i(dP/dz).

3. Mar 31, 2007

### Alesak

thats very clever, thanks. before i even didnt know this rule exists