Math Final Exam.

Transcription

1 Math Final Exam. This is a closed book exam. No calculators are allowed. The exam consists of 8 questions worth 100 points. Good luck! Name: Acknowledgment and acceptance of honor code: Signature: Problem Problem 1 Problem 2 Problem 3 Problem 4 Problem 5 Problem 6 Problem 7 Problem 8 Total Score 1

2 Problem 1 (10 points) Consider the complex number z = e 2πi 5. (i) [2 points] Show that z 4 + z 3 + z 2 + z + 1 = 0. (ii) [2 points] Use (i) to prove that ( z z 2 ) + ( z + 1 z ) + 1 = 0. (iii) [2 points] Let w = z + 1 z. Show that w2 2 = z z 2.

3 (iv) [2 points] Use (ii) and (iii) to determine the possible values of w. (v) [2 points] Explain why w = 2 cos ( ) ( 2π 5. Determine the value of cos 2π ) 5. Problem 2 (12 points) Let u and v be two harmonic functions such that: v is the harmonic conjugate of u, 2u + v = x 2 + 4xy + y 2 + x + 3y u(0) = 1, v(0) = 2. (i) [4 points] Show that u + 2v is the harmonic conjugate of 2u + v.

4 (ii) [4 points] What are the harmonic conjugates of the function x 2 + 4xy + y 2 + x + 3y? (iii) [5 points] Use (i) and (ii) to determine the functions u and v. Be careful about the initial conditions!

5 Problem 3 (12 points) (i) [4 points] Find the different Laurent expansions in powers of z, also indicating where they hold, for the function f(z) = z z 2 4. Please make sure that your final answer shows the three non-zero lowest terms.

6 (ii) [4 points] Find the residue at z = 1 i of the function f(z) = z Log z (z i) 2. (iii) [4 points] For what value of the parameter a, does the function 1 f(z) = e 2z 1 + a sin z have a removable singularity at the origin? For the parameter a you found, how would you define the function at 0 (so that it becomes holomorphic)?

7 Problem 4 (10 points) Compute following the steps below. 2π 0 dθ cos θ (i) [3 points] Write z = e iθ. Express cos θ in terms of z and z 1. Show that dθ = 1 i dz z. (ii) [3 points] Rewrite the integral above as a complex integral in z. (iii) [4 points] Evaluate the integral you obtained using residues.

8 Problem 5 (15 points) Always true or sometimes false? You do not need to indicate the reasoning. T F If f and g have simple poles at 0, then fg has a simple pole at 0. T F If f has an essential singularity at 0, and g has a pole at 0, then f + g has an essential singularity at 0. T F If f is holomorphic in Ω, and C is a simple closed curve contained in Ω, then f(z)dz = 0. C T F Any entire function is the complex derivative of another entire function. T F If f never equals 0 in some region Ω, we can define a holomorphic branch of log f(z). T F At a removable singularity the residue is 0. T F Any function holomorphic in C \ {0} is the derivative of another holomorphic function. T F If f has poles in the region enclosed by a simple close curve C, then C f (z)dz may be non-zero. T F If f has a pole of order m at 0, then f(z 2 ) has a pole of order 2m at 0. T F If f has an essential singularity at 0, and g has a pole at 0, then fg has an essential singularity at 0. T F If f is entire and C 1, C 2 are two simple paths oriented counterclockwise with the same endpoints, then C 1 f(z)dz = C 2 f(z)dz. T F exp ( z + 1 z ) is a meromorphic function. T F The Taylor expansion of a function f around 0 is valid everywhere f is holomorphic. T F The Laurent principal parts of meromorphic functions near the singularities have finitely many nonzero terms. T F The function z sin z e z 1 has a pole at z = 0.

9 Problem 6 (20 points) (i) [10 points] Using residues, evaluate the integral Clearly explain all the necessary estimates. 0 dx x 4 + 3x

10 (ii) [10 points] Using residues, evaluate the integral Clearly explain all the necessary estimates. x sin x x dx.

11 Problem 7 (13 points) Consider the polynomial P (z) = 2z 5 + 6z 3 1. (i) [4 points] How many zeros does P have inside the disc z < 1? (ii) [3 points] Show that all five zeros of P are inside the disc z < 2.

12 1 (iii) [3 points] Consider the Laurent expansion of in powers of z, for z large. For what 2z 5 +6z 3 1 z does this Laurent expansion converges for sure? Explain why 1 2z 5 + 6z 3 1 = 1 2z 5 3 2z 7 + higher order terms in 1 z (iv) [3 points] Determine z =4 dz 2z 5 + 6z 3 1.

13 Problem 8 (8 points) Let f be non-constant holomorphic in the closed unit disc z 1, such that f = 1 on the boundary unit circle z = 1. Follow the steps below to show that f has a zero inside the unit disc. We argue by contradiction, assuming that f is never zero in the unit disc. (i) [2 points] Explain why it follows that f(z) < 1 everywhere inside the unit disc. (ii) [2 points] Consider the function g(z) = 1 f(z). Explain why g(z) < 1 everywhere inside the unit disc. (iii) [2 points] Put (i) and (ii) together to derive a contradiction!

14 (iv) [2 points] Let F (z) = 2z 1 z 2. Confirm that F (z) = 1 on the boundary unit circle, that F is holomorphic in the closed unit disc, and that F indeed has a zero inside the unit disc. Hint: To check that F (z) = 1 you need to verify that 2z 1 = z 2 when z = 1.