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Homework 10 (MATH 1199 Fall 2019)
1. Find a Laurent series, centered at 0, for the function f(z)=1+5z3z4+z7. Use that Laurent series with the Laurent series theorem to compute C1+5z3z4+z7dz where C is the circle |z|=12, oriented positively.

2. Find a Laurent series, centered at 0, for e1z4. Use that Laurent series with the Laurent series theorem to compute Ce1z4dz where C is the unit circle, oriented positively.

3. Find the three series for the function f(z)=1(z2)(z3) in the relevant disks and annuli.

4. Find the residue at z=0 of...
(a) 1z+z2
(b) zcos(1z) (recall: cos(z)=k=0(1)kz2k(2k)! )
(c) zsin(z)z (recall: sin(z)=k=0(1)kz2k+1(2k+1)!)