- #1

freddyfish

- 57

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This is the case - I have solved this and also seen the teacher solve this on the board but I am not satisfied with the method and answer. We used the method of resdues.

This is how my instructor would solve it:

F(cosθ,sinθ)=1/(k-cosθ)

F(x,y)=1/(k-x)=P(x,y)/Q(x,y)

Q(x,y)=k-x

x^2+y^2=1 implies Q(x,y)≠0 iff |k| is greater than 1.

-1≤k≤1 implies that Q(x,y)=0 iff x=k.

Indeed this seems legitimate.

He then evaluates the residues for |k| greater than 1 only.

However, when k is not greater than one, the poles lie on the unit circle in the finite/complex plane. In this case the residue method cannot be used and the answer is therefore not complete. What is the solution of this integral without the assumption that |k| is greater than 1?

Thanks