Complex Analysis
Spring 2001
Homework II Solutions
1. With defined to be the least positive zero of cos t, we established in class Wednesday 2 that t eit was onto the first quadrant of the unit circle in C. Use proven properties of the complex exponential
Complex Analysis
Spring 2001
Homework I Due Friday April 13
1. Conway, Chapter 1, section 3, problem 3. 2. The equation az + b + c = 0 can have solution set consisting of a single point, or a z line in the complex plane. (a) Find necessary and sufficient
Complex Analysis
Spring 2001
Homework I Solution
1. Conway, Chapter 1, section 3, problem 3. Describe the set of points satisfying the equation |z - a| - |z + a| = 2c, where c > 0 and a R. To begin, we see from the triangle inequality that 2|a| = | - 2a|
Complex Analysis
Spring 2001
Homework II Due Friday April 20
1. With defined to be the least positive zero of cos t, we established in class Wednesday 2 that t eit was onto the first quadrant of the unit circle in C. Use proven properties of the complex e
Complex Analysis
Spring 2001
Homework III Solutions
1. Conway, chapter 3, section 3, problem 8 If T z =
az + b show that T (R ) = R if cz + d and only if a, b, c, d can be chosen to be real numbers. It is clear that if a, b, c, d are real, then T maps the
Complex Analysis
Spring 2001
Homework IV Due Friday May 11
1. Conway, chapter 4, section 2, problem 10.
Before working on the next problem, be sure to review Prop. 1.6 in Chapter 3. 2. Conway, chapter 4, section 2, problem 13. 3. Conway, chapter 4, sectio
Complex Analysis
Spring 2001
Homework IV Solutions
1. Conway, chapter 4, section 2, problem 10. z2 + 1 Evaluate dz where (t) = reit for t [0, 2] for all possible values of r, z(z 2 + 4) 0 < r < 2 and 2 < r < . Use partial fractions to write 11 3 1 3 1 z2
Complex Analysis
Spring 2001
Homework V Due Friday May 26
1. Conway, chapter 4, section 5, problem 7. 2. Conway, chapter 4, section 5, problem 9. 3. Conway, chapter 4, section 6, problem 6. 4. Conway, chapter 4, section 6, problem 10. 5. Show that an anal
Complex Analysis
Spring 2001
Homework V Solutions
1. Conway, chapter 4, section 5, problem 7. Let (t) = 1 + eit for 0 t 2. Find ( z )n dz for all positive integers n. z-1
2i By Corollary 5.8, this is (n-1)! times the n - 1rst derivative of f (z) = z n eva
Complex Analysis
Spring 2001
Homework VI Due Friday June 1
1. Conway, chapter 5, section 1, problem 1 b,h,i,j. 2. Conway, chapter 5, section 1, problem 4. 3. Conway, chapter 5, section 1, problem 13. 4. Prove that if f is entire with |f (z)| K + m log(1 +
Complex Analysis
Spring 2001
Homework VI Due Friday June 1
1. Conway, chapter 5, section 1, problem 1 b,h,i,j. Determine the nature of the isolated singularity at z = 0 of the following functions. If the function has a pole, find the singular part, for an