Complex variables

[John88]

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.

[studiot]

Are you asking about winding number?

 

A function of a complex variable requires two complex planes, not one.

It is convention to designate these planes xy and vw.

The xy plane or some subset of it (eg your circle or disk) is the domain and the vw plane is the co-domain or range of the mapping.

[John88]

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.

[studiot]

But that will surely depend upon the function, not the domain you pick it from.

 

Say for instance f(z) maps to a single point in the vw plane. How many circuits of the origin is this?

[John88]

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

[studiot]

I'm sorry I should have said it depends on f(z) as well as the domain.

 

This is because functions like sin(z) will pass go several times for some domains and not at all for others.

 

There is a visual method of determining by inspection for common cases, but a bit long winded for a forum.

 

Pages 130 - 133 of Complex Analysis by Stewart and Tall : Cambridge University Press

 

does the job.

[John88]

Thank you very much!

[uncool]

Assuming that f(0) is 0, and assuming that f has no other roots inside the circle (that is, that f(z) is nonzero for any |z| <= 1), and finally, assuming that f is differentiable, then the circle will make a number of laps around the origin equal to the number of derivatives of f that are 0 at 0. For example, if we take the function f(z) = z, the 0th derivative is 0, but the first derivative is 1, so only 1 derivative is 0 - and we get one loop. If we take the function f(z) = z^2, the 0th derivative is 0, the first derivative is 0, and the second derivative is 1 - so we get two loops.

 

More generally, we can allow multiple roots inside the circle; we then get the number of laps by summing up the number of derivatives that are 0 at each of those points. For example, if we take the function f(z) = z^2 - z/2, which is 0 at 0 and 1/2, we can see that only the 0th derivative is 0 at 0 (the first is -1/2), while at 1/2, the 0th derivative is 0, but the first is 1/2, so the total is 2, and the function makes 2 laps around the origin.

=Uncool-

[John88]

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?

[studiot]

What happens if you move the origin?

 

ie use the substitution h = (z-1)

Edit

What aspect of complex analysis are you studying?

 

I am guessing you are starting contour integration and Cauchy's Theorem?

 

You need to carefully distinguish between a path in the complex plane and a function of a complex variable, which is what I described earlier.

 

A path is a line or series of connected lines (curved or straight) in the complex plane.

A contour is a piecewise smooth path. That is it is composed of a finite number of smooth paths.

 

I assume you have done line integrals in real analysis?

Edited June 24, 2013 by studiot

[John88]

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?

[studiot]

 

• 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.

 

This is true, but what do you understand it to mean?

[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.

[studiot]

Have you considered what happens if you parameterize the path?

 

You seem to have some knowledge of complex analysis, but it would help to draw others into the thread if yo utold us what you are studying and at what level.