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
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?