John88

I understand it to mean that the point A is given by z = x + iy i.e. in the xy plane, and f(z) = u(x,y) + iv(x,y) i.e. in the uv plane.

Yes I've done all that sort of stuff. I just thought there might be some theorems I hadn't studied that might help or something. I'm still confused really, I have been given a question. All I am given is this: The position, A, on the complex plane is given by z, and the position, B, on the complex plane is given by a function of z. How many laps does B make about the origin if A goes once along a circle |z| = 1 once?

Could you refer me to a book with this information in? What happens if f(z) is not zero at zero? For instance, how many laps would f(z) = z^5 +1 make about the origin, upon completion of one circuit of the circle |z| = 1?

Thank you very much!

Yes you're right. I just want a general method in order to find the number of circuits (if any) of a given function.

Apologies I have never heard of winding number, though after googling, you could be right. I just want to know in as simple terms as possible, if you go along any simple closed curve a given number of times (for example once around the circle |z| = 1 in the xy plane) how many circuits would a function make about the origin (e.g. f(z) = z^5 + z + 3) in the vw plane.

If you have a circle in the complex plane, say, |z| = 1, and you have a function of z, f(z), after one continuous anticlockwise circuit of the circle, how many laps about the origin are made in relation to f(z)? I just want a general method. I thought you would transform into polar coordinates and increase the angle theta by 2*pi. But I can't get anything from it.