# Derivating polynomial with complex argument

• Alesak
In summary, the conversation discusses finding the partial derivative of a complex polynomial and different methods to do so. One method suggested is using the chain rule, where the derivative with respect to the variable b is equal to i times the derivative with respect to the complex variable z. The person asking the question also mentions trying other forms of complex numbers and using polar form. They also clarify that this is not for homework, but for personal interest.

#### 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). I am 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. I am even not sure if all my thoughts are correct. maybe tommorow in school i´ll figure something out. and its no homework, I am doing it just for fun.

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).

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