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. [1]

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, 4.5 & 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	8.1–8.4	Introduction to complex numbers and their operations. The complex number system 𝐂. The complex plane.	The real number system 𝐑. Fractional powers and roots of real numbers.	Represent complex numbers algebraically in Cartesian form. Represent complex numbers geometrically as points on a plane. Carry out arithmetic operations with complex numbers. Interpret the geometric meaning of addition, subtraction and complex conjugation. Identify the set 𝐂 of complex numbers as a field extending the real number system 𝐑.
13	8.5–8.7	Polar form of complex numbers. Geometric meaning of complex multiplication and division. Powers and roots of complex numbers. De Moivre’s Theorem.	The complex number system 𝐂. The complex plane. Roots and fractional powers of real numbers.	Represent complex numbers in polar form. Algebraically relate the Cartesian and polar forms of a complex number. Use the identities cis(𝜃+ɸ) = cis𝜃∙cisɸ and (cis𝜃)n = cis(n𝜃) (De Moivre's formula) for the complex trigonometric function cis𝜃 = cos𝜃 + i∙sin𝜃 to evaluate products and powers both algebraically and geometrically. Evaluate all n-th roots of a given complex number both in trigonometric and (when possible) in algebraic closed form, and represent them geometrically.
14	8.8–9.2	Roots and factors of polynomials. The Remainder Theorem. Real and complex roots. The Fundamental Theorem of Algebra.	The complex number system 𝐂. Powers and roots of complex numbers. De Moivre’s Theorem. Polynomials: arithmetic operations, long division, and factorizations.	State and prove the Remainder Theorem. Identify roots with linear factors of a polynomial. Factor given simple polynomials into irreducible factors over ℚ, ℝ and ℂ. State the Fundamental Theorem of Algebra. Use the Fundamental Theorem of Algebra to prove that irreducible real polynomials are linear or quadratic.