Difference between revisions of "Math 532: Complex Analysis"
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Students should be familiar with the following concepts. They should know the technical terms, and be able to implement the methods taught in the course to work associated problems, including proving simple results. | Students should be familiar with the following concepts. They should know the technical terms, and be able to implement the methods taught in the course to work associated problems, including proving simple results. | ||
#'''Essential results from a first course''' Review of power series, integration along curves, Gourset theorem, Cauchy's theorem in a disc, Taylor series, Morera's theorem, singularities, residue calculus, Laurent series, argument principle, harmonic functions, maximum modulus principle. | #'''Essential results from a first course''' Review of power series, integration along curves, Gourset theorem, Cauchy's theorem in a disc, Taylor series, Morera's theorem, singularities, residue calculus, Laurent series, argument principle, harmonic functions, maximum modulus principle. | ||
| − | #'''Entire functions''' | + | #'''Entire functions''' Jensen's formula, functions of finite order, Weierstrass infinite products, Hadamard factorization theorem. |
| − | Jensen's formula, functions of finite order, Weierstrass infinite products, Hadamard factorization theorem. | + | #'''The gamma and zeta functions''' Analytic continuation of gamma function, further properties of Γ, functional equation and analytic continuation of zeta function. |
| − | #'''The gamma and zeta functions''' | + | #'''Conformal mappings''' Conformal equivalence, Schwarz lemma, Montel's theorem, Riemann mapping theorem. |
| − | Analytic continuation of gamma function, further properties of Γ, functional equation and analytic continuation of zeta function. | + | #'''Elliptic Functions and Theta Functions''' Liouville theorems, Weierstrass <math>\mathfrak{p}</math> function, the modular group, Eisenstein series, product formula for Jacobi theta function, transformation laws, application to sums of squares. |
| − | #'''Conformal mappings''' | + | |
| − | Conformal equivalence, Schwarz lemma, Montel's theorem, Riemann mapping theorem. | + | |
| − | #'''Elliptic Functions and Theta Functions''' | + | |
| − | Liouville theorems, Weierstrass <math>\mathfrak{p}</math> function, the modular group, Eisenstein series, product formula for Jacobi theta function, transformation laws, application to sums of squares. | + | |
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Revision as of 11:06, 25 May 2010
Contents
Catalog Information
Title
Complex Analysis.
Credit Hours
3
Prerequisite
Math 352 or instructor`s consent.
Description
A second course in complex analysis including the theory of infinite products, gamma and zeta functions, elliptic functions and the Riemann mapping theorem.
Desired Learning Outcomes
Prerequisites
A knowledge of complex analysis at the level of a first course such as Math 352.
Minimal learning outcomes
Students should be familiar with the following concepts. They should know the technical terms, and be able to implement the methods taught in the course to work associated problems, including proving simple results.
- Essential results from a first course Review of power series, integration along curves, Gourset theorem, Cauchy's theorem in a disc, Taylor series, Morera's theorem, singularities, residue calculus, Laurent series, argument principle, harmonic functions, maximum modulus principle.
- Entire functions Jensen's formula, functions of finite order, Weierstrass infinite products, Hadamard factorization theorem.
- The gamma and zeta functions Analytic continuation of gamma function, further properties of Γ, functional equation and analytic continuation of zeta function.
- Conformal mappings Conformal equivalence, Schwarz lemma, Montel's theorem, Riemann mapping theorem.
- Elliptic Functions and Theta Functions Liouville theorems, Weierstrass <math>\mathfrak{p}</math> function, the modular group, Eisenstein series, product formula for Jacobi theta function, transformation laws, application to sums of squares.
