Cauchy Integral Theorem

  • Thread starter moo5003
  • Start date
  • #1
207
0

Homework Statement


For r=1,3,5 compute the following integral:

Integral over alpha (e^(x^2)/(x^2-6x)dx
Alpha(t) = 2+re^(it) from 0 to 2pi



Homework Equations


Cauchy Integral Formula:
f(z) = 1/(2ipi)Integral over Alpha(f(x)/(x-z)dx)



The Attempt at a Solution



For r = 1, the integral is simply 0 since f(x) is analytic over the domain (critical points being 0 and 6).

For r = 3, I'm a little unsure how to proceed. Obviously plugging in alpha and alpha prime doesnt seem like the correct method (simply because the algebra involved is alot) not to mention a nasty integral from 0 to 2pi at the end. I'm assuming I'm supposed to use the cauchy integral theorem (the chapter we are on) in solving the problem.

Questions: How can I apply the cauchy integral theorem to help me solve this?

At first I tried spliting up the domain into seperate chunks so that some would sum to 0 and then I would be left with something that I already knew, though I couldnt really manage. The best application of the cauchy integral theorem that I can think of is taking a f(z) that I know and setting it equal to the integral relation and then somehow deriving the original integral from the chaos... Is this the correct way of going about it?
 

Answers and Replies

  • #2
207
0
Good news, I figured out my problem. You have to use partial fractions to seperate the deonominator so that you can better apply cauchy formula.

e^(x^2)/(x^2-6) = e^36 / (6x-6) - 1/6x and then just go from there depends on r it changes since domain changes.
 

Related Threads on Cauchy Integral Theorem

  • Last Post
Replies
8
Views
1K
  • Last Post
Replies
3
Views
6K
Replies
3
Views
11K
  • Last Post
Replies
1
Views
624
Replies
2
Views
1K
Replies
2
Views
6K
Replies
6
Views
2K
Replies
2
Views
7K
Replies
1
Views
1K
Top