How should I pick the contour to compute the integral
∫a−a1−z√(z−a)(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
1−z√(z−a)(z+a)∼i−iz+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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