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Name: Unknown
Status: N/A
Age: N/A
Location: N/A
Country: N/A
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

where x=real,a=real,n=integer.

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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