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Math 421: Fall 2002 Final Exam - Problems and Solutions, Exams of Mathematics

University of Massachusetts - AmherstMathematics

The final exam for math 421, fall 2002. The exam covers various topics in complex analysis, including laurent series, singularities, contour integrals, and taylor series. Students are required to solve problems related to finding the laurent series of certain functions, evaluating integrals, and determining the type of singularities.

Typology: Exams

Pre 2010

Uploaded on 08/19/2009

koofers-user-oft
koofers-user-oft 🇺🇸

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Math 421 Final Exam Fall 2002
Name:
Solve problem 1 and 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 .
1. (16 points) a) Show that the Laurent series of 1
sin(z), centered at 0, has the form
1
sin(z)=1
z+1
6z+7
360z3+· · · terms of order at least five.(1)
(You can use equality (1) in the subsequent parts, even if you do not derive it).
b) Find the principal part at z= 0 of the function f(z) = 1−z
z5·sin(z)
pf3
pf4
pf5
pf8
pf9
pfa

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Math 421 Final Exam Fall 2002

Name:

9 will not be graded. Please fill in: Please do not grade Problem number^ Solve problem 1 and 7 out of problems 2 to 9.^ If you solve all 9, then problem.

  1. (16 points) a) Show that the Laurent series of (^) sin(^1 z) , centered at 0, has the form sin(^1 z) =^1 z +^16 z^ +^3607 z^3 +^ · · ·^ terms of order at least five.^ (1) (You can use equality (1) in the subsequent parts, even if you do not derive it).

b) Find the principal part at z = 0 of the function f (z) = (^) z (^51) ·^ −sin(^ zz)

c) Find all the singularities ofdetermine their type (isolated, removable, pole of what order, essential). f (z) (given in part b) in the disk {|z| < 4 } and

d) Find the residue at each isolated singularity in D.

  1. (12 points) Compute the integral^ ∫ C^ sin( e 3 zz (^) −) + 1 ez dz, where C is the circle {|z| = 1} traversed counterclockwise.
  1. (12 points) a) Find the Taylor series of the function f (z) = z z^ −+ 1 1 centered at 0 and determine its radius of convergence. Justify your answer.

b) Find the Laurent series of the functiondomain |z| > 1. f (z), given in part a), valid in the

  1. (12 points) Evaluate the integral ∫ (^2) π 0 5 + 4 sin(^ dθ θ).
  1. (12 points) Let is a positive real number. Prove the inequality CA be the straight line segment from A + iA to −A + iA, where A ∣∣∣ ∣

CA

eiz z^2 + 1dz

∣ ≤^2 Ae −A A^2 − 1.

c) Ifby C C, then there is a single valued branch of log( is a simple closed contour, and z 0 does not belong to the domainz − z 0 ), defined for all D z bounded in D.

d) There exists an entire function, whose real part is xey.

  1. (12 points) Evaluate the improper integral ∫ ∞ 0

dx x^4 + 1