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Math 6322 Function of One Complex Variable: Take-Home Final Exam Solutions, Exams of Mathematics

Avinashilingam UniversityMathematics

Solutions to the take-home final exam for the university of houston math 6322 function of one complex variable course, taught by dr. Min ru in fall, 2011. The exam covers major theorems such as the argument principle, cauchy's integral formula, schwarz-pick theorem, and rouche's theorem, among others. Students are required to prove these theorems and apply them to various problems.

Typology: Exams

2012/2013

Uploaded on 02/12/2013

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Math 6322 Function of one complex variable, Take Home Final Exam
Fall, 2011, Dr. Min Ru, University of Houston
Name(Printed):
Student ID:
Warning: The honor code applies to this exam. You must work alone. You
may consult your notes, books, web, or any other material, but you are not
allowed to talk to or work with anyone else. If you have any question, please
e-mail me. The due date is Wed., Dec 7, before noon.
Here is our list of major theorems (not complete) from the semester (in
alphabetical order).
•Argument principle (for meromorphic functions)
•Big Picard Theorem
•Bloch’s theorem (about the range of holomorphic functions)
•Casorati-Weierstrass theorem
•Cauchys inequalities and estimates for derivatives
•Cauchys integral formula
•Cauchys integral theorem
•Goursat’s theorem
•Hurwitzs theorem
•Laurent expansion (theorem) for holomorphic functions on an annulus
1
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Download Math 6322 Function of One Complex Variable: Take-Home Final Exam Solutions and more Exams Mathematics in PDF only on Docsity!

Math 6322 Function of one complex variable, Take Home Final Exam Fall, 2011, Dr. Min Ru, University of Houston

Name(Printed):

Student ID:

Warning: The honor code applies to this exam. You must work alone. You may consult your notes, books, web, or any other material, but you are not allowed to talk to or work with anyone else. If you have any question, please e-mail me. The due date is Wed., Dec 7, before noon.

Here is our list of major theorems (not complete) from the semester (in alphabetical order).

  • Argument principle (for meromorphic functions)
  • Big Picard Theorem
  • Bloch’s theorem (about the range of holomorphic functions)
  • Casorati-Weierstrass theorem
  • Cauchys inequalities and estimates for derivatives
  • Cauchys integral formula
  • Cauchys integral theorem
  • Goursat’s theorem
  • Hurwitzs theorem
  • Laurent expansion (theorem) for holomorphic functions on an annulus
  • Liouvilles theorem
  • Little Picard Theorem
  • Maximum modulus theorem
  • Marty’s theorem about normal families
  • Montel’s theorem about normal families
  • Moreras theorem
  • Open mapping theorem
  • Power series expansion (theorem) for holomorphic functions
  • Residue theorem
  • Riemann mapping theorem
  • Riemanns removable singularity theorem
  • Rouche’s theorem
  • Schwarz Lemma
  • Schwarz-Pick Theorem
  • Uniqueness theorem (Corollary 3.6.3)

Part A: (1) State and prove the residue theorem (P. 123) and derive, from the residue theorem, the Cauchy’s integral formula (simple version on the disk), and the Argument principle (for meromorphic functions) (i.e. The- orem 5.1.4 on Page 161). (2) Use Cauchy’s integral formula to prove the Laurent expansion (theorem) for holomorphic functions on an annulus. (3) Prove the Schwarz-Pick theorem and use it (or Schwarz lemma) to determine Aut(D(0, 1)), where Aut(D(0, 1)) = {f | f : D(0, 1) → D(0, 1) biholomorphic} (note that Aut(D(0, 1)) forms a group under composition). See P. 182-184 for detail. (4) State the Rouche’s theorem and use it to prove the open mapping