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I had a question;

What is the idea (in complex analysis) of integrating along a square? Take a look at @M.N.C.E.'s method on Evaluate $\int^1_0 \log^2(1-x) \log^2(x) \, dx$

I am not quite sure what the person means by integrating along an infinitely large square?

Can someone tell me where I can find more information about this? I don't have a lot of experience with complex analysis, and would like to learn about this method of squares etc.. and residues.

Help is appreciated!

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  • $\begingroup$ It's not "the method of squares". That's what he chose as a contour. $\endgroup$ – UserX Nov 10 '14 at 17:28
  • $\begingroup$ you choose an contour of integration which looks like a square and let the length of two or all four of it edges go to infinity. The second choice is in many cases equivalent to choose a infinitively large (semi)circle. $\endgroup$ – tired Nov 10 '14 at 17:38
  • $\begingroup$ The easiest ways to learn complex analysis are probably: by a course, by a book, or by a video series. (I don't know if multivariable calculus is a prerequisite; I don't know complex analysis yet.) $\endgroup$ – Akiva Weinberger Nov 10 '14 at 17:47
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You could start here with the Residue Theorem. The theorem relates the path integral of an analytic function with the sum of its residues. Now if you know that the integral is zero (e.g. because your integrand goes to zero), then you also know that the sum of the residues is zero. And that was used in the mentioned answer.
That said, I recommend starting complex analysis from the beginning (most books on the subject will do).

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