Loading [MathJax]/jax/output/HTML-CSS/jax.js
Back to the class
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(z−2)(z−3) 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) z−sin(z)z (recall: sin(z)=∞∑k=0(−1)kz2k+1(2k+1)!)