MATH 153 Final: MATH 153 Harvard Final Exam153 Fall 04
Document Summary
Math 113 practice problems for the final. May, 2005 (1) let f (z) = ez (z 2)(z 4). Find all poles of f and their residues. Compute the integral for r = 1, 4, and 30. Find a fractional linear transformation f taking u to an angular sector, and such that f (0) = 1. (4) evaluate the following integral using the method of residue calculus, for a > 0. Z sinx x(x2 + a2) dx. (5) compute the laurent series for f (z) = z+1 z 1 on (a) the annulus {z | 0 < |z| < 1} (b) the annulus {z | 1 < |z| < } Evaluate res(f ; 0) and res(f ; ). (6) suppose that f = u + iv is analytic on the unit disk. Show that u(0) = for all 0 < r < 1.
