Question #231357

18) Evaluate ∮(z)=1 Z² Sin1/z dZ?

Expert's answer

On the unit circle,"z = sin(i\u03b8)\\space and \\space dz = ie^{i\u03b8} d\u03b8" . We then have

"\\oint|z|=1 z + 1 z2 dz = Z\\intop^ {2\u03c0} _0 (e\u2212i\u03b8+e\u22122i\u03b8)ie^{i\u03b8} d\u03b8"

"= i Z\\intop^ {2\u03c0} _0 (1+e\u2212i\u03b8) d\u03b8 = 2\u03c0i"

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"

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