Question

18) Evaluate ∮(z)=1 Z² Sin1/z dZ?

Answer 1

On the unit circle,$z = sin(iθ)\space and \space dz = ie^{iθ} dθ$ . We then have

$\oint|z|=1 z + 1 z2 dz = Z\intop^ {2π} _0 (e−iθ+e−2iθ)ie^{iθ} dθ$

$= i Z\intop^ {2π} _0 (1+e−iθ) dθ = 2πi$

The integrand $(z + 1)/z^2$ has a double pole at z = 0. The Laurent expansion in a deleted neighborhood of z = 0 is simply $1( z + 1) z^2$ , where the coefficient of 1/z is seen to be 1. We have 

$Res (\frac{z+1}{z^2},0)= 1$