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I'm terribly confused and my advisor is nowhere to be found. Can anyone help with the following scenario?

I am trying to integrate a series of ugly functions of two variables over a solid angle. For most of them, I can get things into a single integral over theta (0 to Pi). The problem is that all of these functions blow up at pi. So, first I integrated and took a limit to try and see if the area underneath the function is finite:

[tex] \lim_{a \rightarrow \pi} \int_0^a f(\theta) d\theta [/tex]

And when I took these limits they all came out to be infinite (not good). But for kicks and because I'd done the algebra already, I tried doing the integrals as contour integrals - made a substitution z=exp(i*theta), etc. and I got nice numerical (finite!!) answers.

I guess this is similar to the integral of [tex] \int_0^1 \frac{1}{x} dx [/tex].

Apparently, if you do this as a contour integral, you get a scalar multiple of pi*i from the residue theorem. But I don't understand this! How can doing the integration one way give you an infinite answer, and the other method give you something finite? What gives? Am I totally wrong on the contour integration, or do you actually get two different results this way? :grumpy:

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# Confusion on contour integration

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