Thread: Complex Contour Integral

I'm attempting to do a complex contour integral as follows:

Integral along C given as the circle |z|=3 in the CCW (positive) direction. Evaluate integral along C of z^-1 + z^-3 dz directly by parameterization of C.

I parameterized:

z(Ɵ)=3e^iƟ 0<=Ɵ<=2*pi

Since integral of f(z)dz along c is integral from a to b of f(z(Ɵ))z'(Ɵ)dƟ by (reasoning? also, do I need to show this is piecewise continuous or is it already taken care of?)

I then integrate [1/(3e^iƟ)+1/((3e^iƟ)^3)]3ie^iƟ dƟ
=integral from 0 to 2pi i+i/9e^2iƟ dƟ

iƟ + -1/18*i*e^-2iƟ

which when evaluated from 0 to 2pi gives me

[2pi*i - 1/18 * i * 1] - [0- 1/18*i]

= 2 * i * pi for my final answer.

(since e^-2pi*i = e^0= 1)


My question, then, is what logical/reasoning steps have I left out and did I get this right? These integrals not done with the residue method really mess me up.

Thanks in advance.

Originally Posted by Negativ

I'm attempting to do a complex contour integral as follows:

Integral along C given as the circle |z|=3 in the CCW (positive) direction. Evaluate integral along C of z^-1 + z^-3 dz directly by parameterization of C.

I parameterized:

z(Ɵ)=3e^iƟ 0<=Ɵ<=2*pi

Since integral of f(z)dz along c is integral from a to b of f(z(Ɵ))z'(Ɵ)dƟ by (reasoning? also, do I need to show this is piecewise continuous or is it already taken care of?)

From a to b? Not really sure what a to b is, but I assume you mean . You can think of z as a function on theta, or you can just think of it as a change of variables. In other words, if .

Originally Posted by Negativ

I then integrate [1/(3e^iƟ)+1/((3e^iƟ)^3)]3ie^iƟ dƟ
=integral from 0 to 2pi i+i/9e^2iƟ dƟ

iƟ + -1/18*i*e^-2iƟ

which when evaluated from 0 to 2pi gives me

[2pi*i - 1/18 * i * 1] - [0- 1/18*i]

= 2 * i * pi for my final answer.

(since e^-2pi*i = e^0= 1)

Okay. I believe you. I couldn't read any of that because of the lack of latex input.

Originally Posted by Negativ

My question, then, is what logical/reasoning steps have I left out and did I get this right? These integrals not done with the residue method really mess me up.

Thanks in advance.

Again, really not sure what you wrote above. But, I can assure you that

Thank you very much for the response. What you interpreted from my spastic text was what I was saying.

Edit: How would one type in the more easily readable text?

Originally Posted by Negativ

Thank you very much for the response. What you interpreted from my spastic text was what I was saying.

Edit: How would one type in the more easily readable text?

Write in latex code, and wrap the code in [ MATH ] and [ /MATH ] tags.

LaTeX:Symbols - AoPSWiki