Ask A Scientist Mathematics Archive

Question: 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
 where x=real,a=real,n=integer.
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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?
Arthur Smith
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