When constructing appropriate contours, we would like it so that the singularities are not on the contour but rather inside or outside the contour.
I see that the integrand has a removeable singularity at z=0. Does this matter? I feel like even if we define f(z)=1, then f is analytic at z=0, so it is fine to construct an integral passing through z=0, (and it will never be fine to construct one passing through ±i.)
Answer
Yes, you are right. If we definef(z)={sinzz if z≠01 otherwise,then your integral is∫+∞−∞f(x)x2+1dx.Since f is an analytic function, you can use the standard methods of Complex Analysis.
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