Answer in Chemical Engineering for pavani #231357
March 11th, 2023
18) Evaluate ∮(z)=1 Z² Sin1/z dZ?
On the unit circle,“z = sin(iu03b8)space and space dz = ie^{iu03b8} du03b8” . We then have
“oint|z|=1 z + 1 z2 dz = Zintop^ {2u03c0} _0 (eu2212iu03b8+eu22122iu03b8)ie^{iu03b8} du03b8”
“= i Zintop^ {2u03c0} _0 (1+eu2212iu03b8) du03b8 = 2u03c0i”
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”
