# MAT3223

## Course Catalog

MAT 3223. Complex Variables. (3-0) 3 Credit Hours.

Prerequisites: MAT 2214 and MAT 3213. An introduction to complex variables, including elementary functions, line integrals, power series, residues and poles, and conformal mappings. Generally offered: Spring. Differential Tuition: \$150.

Textbook: John M. Howie, Complex Analysis, Springer Undergraduate Mathematics Series, Springer-Verlag London (2003). ISBN: 978-1-4471-0027-0. 

Week Sections Topics Student Learning Outcomes
1

2.1, 2.2

Introduction to complex numbers, their operations and geometry.

• Complex numbers and the complex plane.
• Elementary operations on complex numbers (addition, subtraction, multiplication, division, conjugation, modulus, argument).
• Complex numbers in Cartesian and polar forms.
• Complex operations: Elementary algebraic identities and inequalities.
• Geometric meaning of complex arithmetic operations.
• DeMoivre's Formula.

2

3.1, 3.2, 3.3

Topology of the complex plane. Continuous complex functions.

• Essential analysis concepts: sequences, series, limits, convergence, completeness.
• Basic topology of the complex plane: open, closed and punctured discs, open and closed sets, neighborhoods.
• Continuous functions and operations on them.

3

4.1

Complex differentiation

• Definition of complex derivative at a point.
• Cauchy-Riemann equations.
• Examples of differentiable and non-differentiable complex functions.
• Holomorphic functions.

4

4.2

Examples of power series and their formal manipulation.

• Review of Taylor coefficients and Taylor series. Radius of convergence.
• Power series of rational functions.
• Power series defining the complex exponential, trigonometric and hyperbolic functions.

5

4.3, 4.4 & 4.5

Complex natural logarithms. Multivalued holomorphic functions. Singularities.

• Definition of the multivalued complex natural logarithm, its principal branch, and other branches.
• Complex powers via logarithms.
• Definition of branch point and branches.
• Functions holomorphic in punctured neighborhoods. Poles and other singularities.
• Examples of branching and singularities (complex logarithms, inverse trigonometric/hyperbolic functions, and complex powers).

6

None

Review. First midterm exam.

7

5.2 & 5.3

Parametric curves. Line integrals.

• Parametric representation of piecewise smooth curves.
• Arc-length. Rectifiable curves.
• Line integrals: Definition, examples, and elementary properties.
• Line integrals of holomorphic functions. Fundamental Theorem.

8

5.4 & 5.5

Estimation and convergence of line integrals.

• Majorization of path integrals by arclength and bound on magnitude of integrand.
• Antiderivatives of complex functions with path-independent line integrals.
• Uniform and non-uniform convergence of sequences and series of complex functions.
• Continuous uniform limits of continuous sequences and series, and their integrals.

9

6.1, 6.2, 6.3

Cauchy's Theorem and its basic consequences.

• Statement of Cauchy's Theorem.
• Proof of Cauchy's Theorem.
• The Deformation Theorem.

10

7.1 & 7.2

Cauchy's Integral Formula. Taylor series.

• Statement and proof of Cauchy's Integral Formula.
• Existence, uniqueness, and general theory of Taylor series of holomorphic functions.
• Rigorous definition of and proof that complex logarithms are holomorphic.

11

None

Review. Second midterm exam.

12

8.1–8.3

Isolated singularities and Laurent series. The Residue Theorem.

• Definition of Laurent series about an isolated singularity. Examples.
• Types of isolated singularities: Removable, polar, essential. The Cassorati-Weierstrass Theorem.
• Statement and proof of the Residue Theorem.
• Elementary techniques to evaluate residues.

13

Chapter 9.

Calculus of residues.

• Evaluation of integrals of real analytic functions using residues.
• Evaluation of series of real analytic functions using residues.

14

11.1–11.3

Conformal mappings.

• Preservation of angles and conformal mappings of the plane.
• Conformal mappings yield pairs of conjugate harmonic functions.
• Dirichlet's Problem on a planar region.
• The Riemann Mapping Theorem.
• Möbius transformations and their use in solving elementary Dirichlet Problems.

15

Chapter 10. (At instructor's discretion, week 15 may be used to wrap-up and review instead.)

Complex integration and geometric properties of holomorphic functions

• Rouché's Theorem.
• The Open Mapping Theorem.
• Winding numbers.