Sunday, 30 December 2018

Complex integration: choosing a contour without divergences

How should I pick the contour to compute the integral
aa1z(za)(z+a)dz,
where a is a real number?




My problem is that when I choose a keyhole contour around the cut (a,a), the big circle with radius R going to infinity diverges. The integrand goes as
1z(za)(z+a)iiz+ia22z2+O(1z3),
for |z| and thus
lim



But I know that the answer is finite as



\int_{-a}^a\frac{1-z}{\sqrt{(z-a)(z+a)}}\mathrm d z =\pi\,.



Where am I doing something wrong?

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