Contour integration with a branch cut

[mercenarycor]

Homework Statement

∫_-1¹ dx/(√(1-x²)(a+bx)) a>b>0

Homework Equations

f(z₀)=(1/2πi)∫f(z)dz/(z-z₀)

The Attempt at a Solution

I have absolutely no idea what I'm doing. I'm taking Mathematical Methods, and this chapter is making absolutely no sense to me. I understand enough to tell I'm supposed to do contour integration on this with a branch cut on the singularity, but actually doing it is another thing. Also, I have no idea what to do with the second term in the denominator. If you can explain this to me, I would be grateful; and please, try to dumb it down. I can't even figure out how to find residues.

The farthest I got was K=∫_-1¹dx/(√(1-x²)(a+bx)) + lim _r->∞∫₀^π re^iθidθ/(√(1-r²e^i2θ)(a+bre^iθ))
I stopped there, however, because I'm fairly certain I'm embarking on several hours of barking up the wrong tree.

 

[strangerep]

Have you had a proper course, or part-course, on contour integration? Without that, this will be very difficult.

I could tell you that you're supposed to place a branch cut between $x = \pm 1$, and use a "dog bone contour" (aka "dumbbell contour"), but that won't be much help if you don't know how to do easier contour integrals and compute basic residues. [Google for "dog bone contour" to see what this looks like.]

 

[vela]

Have you had a proper course, or part-course, on contour integration? Without that, this will be very difficult.

The OP is taking a math methods course right now and learning about contour integration and complex analysis right now.

 

[vela]

I have absolutely no idea what I'm doing. I'm taking Mathematical Methods, and this chapter is making absolutely no sense to me. I understand enough to tell I'm supposed to do contour integration on this with a branch cut on the singularity, but actually doing it is another thing. Also, I have no idea what to do with the second term in the denominator. If you can explain this to me, I would be grateful; and please, try to dumb it down. I can't even figure out how to find residues.

As strangerep suggested, you should probably go back to trying to do easier problems first and getting a handle on those before trying to tackle this one.

You could try finding a similar but simpler example in your textbook (one singularity, single branch cut) and asking questions about that first.