MATH 3364 - Introduction to Complex Analysis - University of Houston

# MATH 3364 - Introduction to Complex Analysis

***This is a course guideline.  Students should contact instructor for the updated information on current course syllabus, textbooks, and course content***

Prerequisites: MATH 3331.

Course Description: The complex number system, analytic functions, the Cauchy integral theorem, series representation, residue theory, and conformal mapping

Text: Fundamentals of Complex Analysis with Applications to Engineering and Science, 3rd Edition, by E.B. Saff and A.D. Snider, Prentice-Hall, 2003. ISBN: 9780139078743

Syllabus

Chapter 1: Complex Numbers
1.1 The Algebra of Complex Numbers
1.2 Point Representation of Complex Numbers
1.3 Vectors and Polar Forms
1.4 The Complex Exponential
1.5 Powers and Roots
1.6 Planar Sets
1.7 The Riemann Sphere and Stereographic Projection

Chapter 2: Analytic Functions
2.1 Functions of a Complex Variable
2.2 Limits and Continuity
2.3 Analyticity
2.4 The Cauchy-Riemann Equations
2.5 Harmonic Functions

Chapter 3: Elementary Functions
3.1 Polynomials and Rational Functions
3.2 The Exponential, Trigonometric and Hyperbolic Functions
3.3 The Logarithmic Function
3.4 Washers, Wedges, and Walls
3.5 Complex Powers and Inverse Trigonometric Functions

Chapter 4: Complex Integration
4.1 Contours
4.2 Contour Integrals
4.3 Independence of Path
4.4 Cauchy's Integral Theorem
4.5 Cauchy's Integral Formula and Its Consequences
4.6 Bounds for Analytic Functions

Chapter 5: Series Representations for Analytic Functions
5.1 Sequences and Series
5.2 Taylor Series
5.3 Power Series
5.4 Mathematical Theory of Convergence
5.5 Laurent Series
5.6 Zeros and Singularities
5.7 The Point at Infinity

Chapter 6: Residue Theory
6.1 The Residue Theorem
6.2 Trigonometric Integrals
6.3 Improper Integrals of Certain Functions
6.4 Improper Integrals Involving Trigonometric Functions
6.5 Indented Contours
6.6 Integrals Involving Multiple-Value Functions
6.7 The Argument Principle and Rouche's Theorem
At the instructor's discretion, other topics as time permits.

CSD Accommodations:

Accommodation Forms: Students seeking academic adjustments/auxiliary aids must, in a timely manner (usually at the beginning of the semester), provide their instructor with a current Student Accommodation Form (SAF) (paper copy or online version, as appropriate) from the CSD office before an approved accommodation can be implemented.

Details of this policy, and the corresponding responsibilities of the student are outlined in The Student Academic Adjustments/Auxiliary Aids Policy (01.D.09) document under [STEP 4: Student Submission (5.4.1 & 5.4.2), Page 6]. For more information please visit the Center for Students with Disabilities Student Resources page.

Additionally, if a student is requesting a (CSD approved) testing accommodation, then the student will also complete a Request for Individualized Testing Accommodations (RITA) paper form to arrange for tests to be administered at the CSD office. CSD suggests that the student meet with their instructor during office hours and/or make an appointment to complete the RITA form to ensure confidentiality.

*Note: RITA forms must be completed at least 48 hours in advance of the original test date. Please consult your counselor ahead of time to ensure that your tests are scheduled in a timely manner. Please keep in mind that if you run over the agreed upon time limit for your exam, you will be penalized in proportion to the amount of extra time taken.

UH CAPS

Counseling and Psychological Services (CAPS) can help students who are having difficulties managing stress, adjusting to college, or feeling sad and hopeless. You can reach (CAPS) by calling 713-743-5454 during and after business hours for routine appointments or if you or someone you know is in crisis. No appointment is necessary for the "Let's Talk" program, a drop-in consultation service at convenient locations and hours around campus.