Spring 2010 Exam 2. You may not use your books, notes, or any calculator on this exam.

Transcription

1 MTH 282 final Spring 2010 Exam 2 Time Limit: 110 Minutes Name (Print): Instructor: Prof. Houhong Fan This exam contains 6 pages (including this cover page) and 5 problems. Check to see if any pages are missing. Enter all requested information on the top of this page, and put your initials on the top of every page, in case the pages become separated. You may not use your books, notes, or any calculator on this exam.

2 MTH 282 final Exam 2 - Page 2 of 6 1. (20 points) Concepts: Only write T(rue) or F(alse) for each propostion. You don t need to give any argument. 1. Suppose f(z) is a complex-valued function defined on a nonempty open set D of C. Then f (z) exists at a C if and only if f(z) is differentiable as a real function of two variables and the Cauchy-Riemann equations are satisfied. 2. Suppose f(z) is a complex-valued function defined on a nonempty connected open set D of C. Let u denote the real part of f(z) and let v denote the imaginary part. For any z D, we have sin 3 u + cos 3 v = 11. Then f(z) is a constant over D. 3. Suppose f(z) is a complex-valued function defined on a nonempty open set D of C. a D. Then the radius of convergence of the Taylor series of f(z) at a is greater than or equal to the radius of the largest open disc in D centering at a. 4. If the radius of convergence of the power series a n z n is r > 0 and b is a point in C with modulus r. If the power series converges at b, then you still may not find an analytic continuation of the power series at b. On the other hand, if you can find an analytic continuation of the power series at b, you still can t claim the power series converges at b. 5. Suppose f(z) is a nonzero continuous complex-valued function defined on a nonempty open set D of C. Then the multiple-valued function Arg(f(z)) has a monodromy branch in D. 6.(For this problem, you need to write down the answer instead of judging whether it is correct) Write a MAXIMAL open subset of C such that there is a monodromy analytic function f(z) such that f 2 (z) = z(z 1)(z 2). 7. Suppose f(z) is an analytic function defined on {z z C, z > 1}, and is a removable singularity of f(z), then the residue of f(z) at is Suppose f(z) is a complex-valued function defined on a nonempty open set D of C. a D. f(z) is analytic on D {a}. a is a removable singularity of f(z). Then f(z) is a meromorphic function on D 9. Suppose f(z) is a complex-valued function defined on a nonempty open set D of C. f (a) exists. Then f(z) is analytic at a. 10. If the radius r of convergence of the power series a n z n is greater than 0. b C, b = r. lim b. z <r,z b n=0 a n z n does not exist, then a n z n do not have an analytic continuation at n=0 n=0 n=0

3 MTH 282 final Exam 2 - Page 3 of 6 2. (20 points) Construct an one-to-one conformal mapping from the domain {z C z 1 < 2, z + 1 < 2} to the domain {z C 0 < Imz < π}.

4 MTH 282 final Exam 2 - Page 4 of 6 3. (20 points) Let f(z) = sin( z z 1 ). (a) (5 points) Point out all the singularities of f(z) in C (b) (5 points) Give the type of each singularity. (c) (10 points) Compute the residue of f(z) at each singularity

5 MTH 282 final Exam 2 - Page 5 of 6 4. (20 points) Let f(z) = sin( z z 1 ). (a) (8 points) How many roots of 7z 3 x 2 + z + 1 = 0 are lying inside the disk z < 1? (b) (12 points) We know the general solution to the complex ordinary differential equation y + 2y + 3y + y = 0 is of the form y = c 1 e λ 1t + c 2 e λ 2t + c 3 e λ 3t where c 1, c 2, c 3 are three arbitrary complex constant and λ 1, λ 2, λ 3 are the three roots of 7z 3 x 2 +z +1 = 0. Prove: lim t R,t + y(t) = 0

6 MTH 282 final Exam 2 - Page 6 of 6 5. (20 points) Give all the complex-valued funtions V (z) satisfying the following properties: (1) V (z) is analytic on Ω = z z C, z > 1. (2) V (z) is real continuous on overlingω. (3) V ( ) = c for a given complex constant c. (4) On the unit circle Ω = z z C, z = 1, V (z) is tangent to the circle.