Integral Solution Residues; Residue Theorem
Date: Around 1993
Can you solve the following integral from 0 to 2*pi without using
the residue theorem?
(sin(x) * exp(-i*n*x))/(a + cos(x)) dx
Well, for specific values of n, you can easily solve it by
substituting y = cos(x). The problem is that e^(-inx) just cries out for
substitution of z = e^ix or some such thing, if you want to do it for general
n, which leads you into complex analysis and the residue theorem. If you can
figure out how to solve an integral of (x - i sqrt(1 - x^2))^n in general,
more power to you, but what is wrong with using residues?
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Update: June 2012