# Contour integration with a branch cut

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1. Feb 28, 2017

### mercenarycor

1. The problem statement, all variables and given/known data
-11 dx/(√(1-x2)(a+bx)) a>b>0

2. Relevant equations
f(z0)=(1/2πi)∫f(z)dz/(z-z0)

3. 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->∞0π reidθ/(√(1-r2ei2θ)(a+bre))
I stopped there, however, because I'm fairly certain I'm embarking on several hours of barking up the wrong tree.

2. Feb 28, 2017

### 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.]

3. Mar 1, 2017

### vela

Staff Emeritus

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

4. Mar 1, 2017

### vela

Staff Emeritus

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.