Complex Variables integration formulas

In summary, the homework statement says that if z. equals zero, then R=1 and the integral of (1/z)*dz comes out to 2i*pi. If z. does not equal zero, then R=0 and the integral of (1/z)*dz comes out to e^(ipi).
  • #1
MadCow999
5
0

Homework Statement


Let C. denote the circle |z-z.|= R, taken counter clockwise. use the parametric representation z= z. + Re^(io) (-pi </= o </= pi) for C. to derive the following integration formulas:
integral C. (dz/(z-z.)) = 2ipi


Homework Equations


note: z. and C. represent z knot and c knot , and </= represents less than or equal to
and o represents theta

The Attempt at a Solution



i was able to do some work to it, but i eventually came out with (integral of) (R i e^io)/R which goes to integral of (i e ^(io))from -pi to pi, which comes out to e^(ipi) - e^(-ipi), which my teacher showed me comes out to zero. (i didnt quite understand his method, but i do know 0=/=2ipi
T_T
thanks!
 
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  • #2
It's a little hard to follow your notation. But for simplicity put 'z.'=0 and R=1. Then you are integrating (1/z)*dz where z=exp(i*theta). You are not integrating exp(i*theta)*d(theta) which it looks like you are doing. exp(i*pi)=(-1). exp(-i*pi)=(-1), so difference is zero. Your teacher was right about that.
 
Last edited:
  • #3
hmm. so it seems to work if 'z.' = 0, but how can i say that? same goes for R = 1
is it because z. is just some arbitrary point i can pick?
we've done integral of (1/z)*dz, where z= R(exp(i*theta)) and dz = R*i(exp(i*theta))d(theta)
that comes out to be 2i*pi, but we were also doing over the interval of (0 to 2pi)

we're supposed to prove it in general i believe, so i would like to try and leave the z. and R in there.

i think I am seeing the relationship now...do i just say let z= z-z. and that yields the same (1/z) integral I've done before?

sorry bout sort of train of thought post...just trying to keep you updated with what I am doing :)
 
  • #4
If you did it with z.=0 and R=1 I think you'll have no problems working it out in general. Yes, you'll get the same thing. You should notice the exponential part and the R just plain cancel. Welcome to PF.
 
  • #5
okiedokie
thanks!
 

1. What are complex variables integration formulas?

Complex variables integration formulas are mathematical equations used to calculate the integration of complex-valued functions. These formulas involve complex numbers, which have both a real and imaginary component, and are used to solve problems in mathematics and science.

2. Why are complex variables integration formulas important?

Complex variables integration formulas are important because they allow us to solve integrals involving complex numbers, which are essential in many areas of physics and engineering. They also help us understand the behavior of complex-valued functions and their relationship to real functions.

3. What is the difference between complex variables integration and regular integration?

The main difference between complex variables integration and regular integration is that complex variables integration involves complex numbers, while regular integration only involves real numbers. In regular integration, the integral is a real-valued function, whereas in complex variables integration, the integral is a complex-valued function.

4. What are some common complex variables integration formulas?

Some common complex variables integration formulas include the Cauchy integral formula, the residue theorem, and the Cauchy-Riemann equations. These formulas are used to solve integrals involving complex numbers and are essential in complex analysis and other areas of mathematics.

5. How can I apply complex variables integration formulas in real-life situations?

Complex variables integration formulas have many practical applications, including in electrical engineering, quantum mechanics, and fluid dynamics. These formulas are used to solve problems involving complex-valued functions, such as calculating the electric field in a circuit or the flow of a fluid around an object.

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