MATH 555 - Introduction to Functions of a Complex Variable with Applications
Fall 2013, Section 001
Instruction Mode: Section 001 is (see other Sections below)
Subject: Mathematics (MATH)
Department: LSA Mathematics
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Details

Credits:
3
Requirements & Distribution:
BS
Waitlist Capacity:
10
Advisory Prerequisites:
MATH 451 or equivalent experience with abstract mathematics.
BS:
This course counts toward the 60 credits of math/science required for a Bachelor of Science degree.
Repeatability:
May not be repeated for credit.
Primary Instructor:

Description

Goals. This course is an introduction to the theory of functions of a complex variable with special attention to applications in science and engineering. In addition to the presentation of the often amazing fundamental principles, the importance and utility of these principles in applications is emphasized. Applications include

  • The fundamental theorem of algebra.
  • Interactions with Fourier analysis, e.g., Shannon’s sampling theorem.
  • The evaluation of definite integrals by the method of residues. Including examples of Fourier and Laplace Transforms.
  • The solutions of boundary value problems for Laplace’s equation arising in fluid mechanics, heat conduction and electrostatics.

Students will be expected to learn to use the methods of complex analysis with facility. Complete proofs will be presented. Students will not be expected to reproduce the harder proofs. That is the province of MATH 596. Compared to MATH 596, product representations, normal families, and the Riemann Mapping Theorem will not be treated. Elementary conformal mappings are treated and applied extensively.

Content. The heart of the course is the derivation of the properties of analytic functions from Cauchy’s Integral Theorem. The latter is proved by Green’s Theorem assuming continuous differentiability. This and the following derivations are simple compared to the proofs in Advanced calculus in several variables or to the theory of the Lebesgue Integral. The results are frequently surprising. Some of the applications are astonishingly elegant and surprising.

Text. Richard A. Silverman, Complex Analysis with Applications, Dover publishers. This required text has the right level, right material, and the right price.

Course Requirements:

Grading. Grades are based on Homework 35%, Midterm Exam 25%, and a Final Exam 40%.

Intended Audience:

Mathematics, science, and engineering undergraduates and graduate students. Prereqs: Advanced Calculus (partial derivatives, Green’s Theorem, uniform convergence and uniform continuity). It is assumed that students have (briefly) encountered Fourier series and Fourier integrals. Many fundamental results will be briefly reviewed when used. MATH 555 is an opportunity to deepen understanding of advanced calculus.

Schedule

MATH 555 - Introduction to Functions of a Complex Variable with Applications
Schedule Listing
001 (LEC)
 In Person
12157
Open
7
 
-
TuTh 1:00PM - 2:30PM

Textbooks/Other Materials

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Syllabi

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