Homework 4 - Complex Analysis - Practice Questions | MAT 572, Assignments of Mathematics

Download Homework 4 - Complex Analysis - Practice Questions | MAT 572 and more Assignments Mathematics in PDF only on Docsity!

MAT 572 Homework 4 Due Wednesday, 10/4 Always prove your answers, 1. Find a Mobius transformation T that carries land |z —1/4| = 1/4 to concentric cirel (Hint: find two points that ai with respect to both circles.) Prove that the ratio of the radii of the concentric circles is independent of the choice of T. 2. Compute Jj w= (z+r?/2)/2 for |2| =r, and using winding mumbers. Jedz in two ways: first by parametrization, then by observing that 3. Compute (Hint: use partial fractions and winding numbers.) 4. Define integration with respect to 7 as follows: =/ F(z) dz. (i1) Prove that if f(z) is a polynomial, then | Fla) dz = —2niR? f(a). Je-a|eK 5. Give an example of a piecewise C+ curve + with the property that for every integer k. there is a complex number a, ¢ {y} such that n(y.a,) =k.