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Complex numbers › Complex analysis › Contour integrals

Contour integral of a square-root branch

University · ●●●○○ 50 pts

Let CC be the circle ∣z∣=R|z|=R with R>1R>1, centred at the origin. Use the branch of (z2−1)1/2(z^2-1)^{1/2} with a straight branch cut joining z=−1z=-1 and z=1z=1, chosen so that (z2−1)1/2>0(z^2-1)^{1/2}>0 for real x>1x>1. Let I=∮C(z2−1)1/2 dzI=\oint_C (z^2-1)^{1/2}\,dz, taken counterclockwise. Write I=b iI=b\,i with bb real. Find bb, rounded to two decimal places.

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contour integrationLaurent seriesresidues