%p22S.txt Solutions to Dr. Z.'s Intro to Complex Variable Attendance Quiz for Lecture 22 1. Explain why it is impossible to have an analytic function f(z) such that f(0)=0, |f(z)|<=9 in the disc |z|<6 and |f(2i)|=4 Sol. to 1: Consider the new function F(z)=f(6z)/9. F(0)=0, F(z) is analytic in |z|<1, and |F(z)|<=1. By Schwarz's lemma: |F(z)|<=|z| for all |z|<1 Hence |F(i/3)|<=|i/3|=1/3 Hence |f(2i)/9| <=1/3 Hence |f(2i)|<=9/3=3. So there is no way that |f(2i)|=4. 2. Compute the following integral: Integral From 0 to 2*Pi of (i+5*exp(it))^10 Sol. to 2: The function featuring in the integral is f(z)=z^10, and it is always analytic so we don't have to worry about the validity, and r=5 is IRRELEVANT (the answer does not depend on r). z_0=i, so the answer is (2*Pi)* f(i)= 2*Pi* i^10= 2*Pi*i^8*i^2= 2*Pi*1*(-1)= -2*Pi Ans. to 2: The value of the integral is -2*Pi