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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. [1]

Week Sections Topics Student Learning Outcomes

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.


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.



Complex differentiation

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



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.


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).



Review. First midterm exam.


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.


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.


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.


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.



Review. Second midterm exam.



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.


Chapter 9.

Calculus of residues.

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



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.


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.