Problem Set 1
Due January 28rd , 2007 1. Evaluate (8 + 2i) (1 i) (2 + i)2 2 3i 8 + i 1 + 2i 6 i
2. Let z be a complex number such that Im z > 0, show that Im(1/z ) < 0. 3. Show that if |z | = 1(z = 1), then Re 1/(1 z ) = 1/2. 4. Find the principal value o
Problem Set 10
Due April 22nd , 2008 1. Prob. 1, 3 on page 336. 2. Prob. 2 on page 344. 3. Prob. 5, 11 on page 345. 4. Prob. 1, 2 on page 354. 5. Show that for a > e, the equation ez = az n has n roots inside the unit disk.
Problem Set 6
Due March 11th , 2008 1. Prob. 3 c), d) and f) on page 212. 2. Prob. 9 on page 213 3. Prob. 15 on page 214 4. Prob. 1 on page 219. 5. Prob. 4 on page 219. 6. Prob. 7 on page 220. 7. Suppose function f (z ) is analytic inside the disk |z | R,
Problem Set 4
Due February 26th , 2007 1. Write the function f (z ) = 2. Verify the identity sin z sin w = 2 cos and show that sin z = sin w if and only if z = w + 2k where k is an integer. 3. Prove that Log ez = z if and only if < Im z < . 4. Find the de
Problem Set 3
Due February 12th , 2008 1. Let g (x + yi) = 3x2 + 2x 3y 2 1 + i(6xy + 2y ). Show that g is an analytic function on C. 2. Show that if f is analytic in a domain D and either Re f (z ) of Im f (z ) is a constant in D, then f (z ) must be a co
Solution: Problem Set 2
1. Consider following sets: a) A = cfw_z : | Arg z | < /4 b) B = cfw_z : 1 < Im z < 1 c) C = cfw_z : |z | 1 d) D = cfw_z : (Re z )2 1 Which of these sets are open? Which are closed? Which are bounded? Which are domains? (No need to
Problem Set 2
Due February 5th , 2008 1. Consider following sets: a) A = cfw_z : | Arg z | < /4 b) B = cfw_z : 1 < Im z < 1 c) C = cfw_z : |z | 1 d) D = cfw_z : (Re z )2 1 Which of these sets are open? Which are closed? Which are bounded? Which are domain
Solution: Problem Set 1
1. Evaluate (8 + 2i) (1 i) (2 + i)2 2 3i 8 + i 1 + 2i 6 i
2. Let z be a complex number such that Im z > 0, show that Im(1/z ) < 0. Proof. Let z = x + yi with both x and y being real numbers, and y > 0. Then x 1 y = 2 2, z |z | |z |