Derivating polynomial with complex argument

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.
  • #1
Alesak
111
0
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.
 
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  • #2
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
thats very clever, thanks. before i even didnt know this rule exists
 

FAQ: Derivating polynomial with complex argument

What is a polynomial with complex argument?

A polynomial with complex argument is a mathematical function that involves complex numbers in its input. It is usually written in the form of f(z), where f is the polynomial function and z is a complex number.

How do you differentiate a polynomial with complex argument?

To differentiate a polynomial with complex argument, you can use the same rules as differentiating a polynomial with real coefficients. The only difference is that you need to treat the complex number z as a variable and use the rules of complex arithmetic to simplify the expression.

What is the purpose of deriving a polynomial with complex argument?

The purpose of deriving a polynomial with complex argument is to find the rate of change of the polynomial function at a specific point in the complex plane. This can help in solving problems related to complex analysis and engineering.

Can you derive a polynomial with complex argument using the chain rule?

Yes, you can use the chain rule to derive a polynomial with complex argument. The chain rule states that if f and g are differentiable functions, then the derivative of the composite function f(g(z)) is given by f'(g(z)) * g'(z).

Are there any special cases when deriving a polynomial with complex argument?

Yes, there are special cases when deriving a polynomial with complex argument. For example, if the polynomial function only has real coefficients, then the derivative will be a real-valued function. Additionally, if the polynomial function is a constant, then its derivative will be zero.

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