Download Math 421 Spring 2005 Final Exam: Problems in Complex Analysis - Prof. Eyal Markman and more Exams Mathematics in PDF only on Docsity!
Math 421 Final Exam Spring 2005
Name:
Solve problem 1 and only 7 out of problems 2 to 9. If you solve all 9, then problem 9 will not be graded. Please fill in: Please do not grade Problem number. Show all your work. Credit will not be given for an answer without a justification. Calculators may not be used in this exam.
- (16 points) Given that the Taylor series of tan(z), centered at 0, has the form
tan(z) = z +
z^3 +
z^5 + · · · terms of order at least seven. (1)
a) Evaluate the fifth derivative tan(5)(0) with as little calculations as possible.
b) Find the principal part at z = 0 of the function f (z) =
(1 + z) tan(z) z^5
c) Find all the singularities of f (z) (given in part b) in the disk D = {|z| < 4 } and determine their type (isolated, removable, pole of what order, essential).
d) Find the residue at each isolated singularity in D.
- (12 points) Compute the integral
C
z^5 1 − z^3
dz, where C is the circle of radius 2, centered at 0, and traversed counterclockwise.
- (12 points) a) Find the Taylor series of the function f (z) =
2 z + 1 z^2 + z − 2
z − 1
z + 2 centered at 0 and determine its radius of convergence. Justify your answer.
b) Find the Laurent series of the function f (z), given in part a), valid in the annulus 1 < |z| < 2.
- (12 points) Evaluate the integral
∫ (^2) π
0
dθ 2 + cos(θ)
Show all your work!
- (12 points) Let SR be the upper-semi-circle of radius R > 1, given by the parametriza- tion z = Reiθ^ , 0 ≤ θ ≤ π. Prove the equality
Rlim→∞
SR
z^2 dz 1 + z^4
Hint: Find first an upper bound for the integral.
c) If f is a non-constant entire function and |f (z)| ≤ 2, for every z on the unit circle {z : |z| = 1}, then f must map the unit disk {z : |z| < 1 } into the disk {z : |z| < 2 }.
d) There exists an entire function, whose real part is ex+y.
- (12 points) Evaluate the improper integral
∫ (^) ∞
0
x^2 x^4 + 1
dx.
