HOMEWORK FOR COMPLEX ANALYSIS STUDENTS

In this document you will find a record of homework assignments.

Monday
Sec. 22:
Exercises: 1, 7(a), 10(c), 11
Wednesday
Whoops! Forgot to give an assignment today!
Friday
Sec. 23:
Exercises: 2, 3, 4
Extra Problem 1: Show that |e^z| is less than or equal to e^|z|.

Monday
Sec. 33:
Exercises: 13(a)(b)
Extra Problem: Find a contour C such that |integral over C of 1/z dz| is not less than or equal to the integral over C of |1/z| dz.
Wednesday
No assignment.
Friday
Sec. 38:
Exercises: 2(a), 5(a)(b), 13

Monday
Problem 1: Assume that f(z) and g(z) are analytic on and inside the simple closed contour C. Assume also that g(z) is not 0 on and inside C. Show that if |f(z)| is less or equal to |g(z)| on C, then this holds inside C as well. Show that if we do not assume that g(z) is not 0 on and inside C, then this conclusion may be false.
Problem 2: Suppose that f(z) is analytic on and inside a simple closed contour C, and that |f(z)-1| is strictly less that 1 on C. Prove that f(z) is never zero inside C.
Problem 3: Suppose that f(z) is a nonconstant function that is analytic on and inside a simple closed contour C. Show that if |f(z)| is constant on C, then f(z) has a zero in C. (Hint: Otherwise both f(z) and 1/f(z) are analytic on and inside C.)
Wednesday
Sec. 44:
Exercises: 1, 3, 4, 5
Friday
Problem 1: Show that if (z_1 + z_2 + z_3 + ...) converges, then lim_{n goes to infinity} z_n = 0.
Problem 2: Assume that lim_{n goes to infinity} |z_{n+1}/z_n| = rho < 1. Show that (z_1 + z_2 + z_3 + ...) is absolutely convergent.

Monday
No assignment.
Wednesday
No assignment.
Friday
Sec. 51:
Exercises: 5, 10, 11(b)
Problem 1: Show that an entire function f(z) is even (f(z) = f(-z)) if and only if f(z) = g(z^2) for some entire function g(z).

Last modified on Aug 23, 2000