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	Power (Taylor) series of holomorphic functions.	Taylor coefficients and Taylor series of a holomorphic function. Radius of convergence. Differentiation of Taylor series. Taylor series of rational functions. The complex exponential, trigonometric and hyperbolic functions and their Taylor series.
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. Derivatives of inverse functions. Derivative of the complex natural logarithm. 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	4.5,	Arithmetic congruences and basic modular arithmetic. Tests of divisibility.	Divisibility of integers. The Division Algorithm.	Use arithmetic congruences to interpret the remainder of integer division. Use congruences to compute remainders of divisions where the quotient is large or irrelevant. Prove basic divisibility criteria by 2, 3, 5, 9 and 11 for number in base 10, using modular arithmetic.
8	3.4	Modular rings ℤₙ. Modular fields ℤₚ. Fermat's Little Theorem.	Primes. Arithmetic congruences and basic modular arithmetic.	Recognize the modular rings ℤₙ as number systems. Evaluate sums, differences, negations and products in ℤₙ. Identify invertible and non-invertible elements of ℤₙ. Find the inverse (when defined) of a given element of ℤₙ. Prove that the modular ring ℤₚ is a field if and only if p is prime. Correctly state Fermat's Little Theorem, both as a theorem in modular arithmetic modulo a prime p, and as a theorem for the finite field ℤₚ. Apply Fermat's Little Theorem to solve arithmetic problems.
9	4.1	Mathematical Induction. Inductive proofs.	Basic proofs. Tautologies and Deductions. Quantifiers. Divisibility of integers.	State the Principle of Mathematical Induction (PMI). Prove elementary algebraic and arithmetic statements by induction. Prove elementary algebraic and arithmetic statements by strong induction.
10	4.2 & 4.3	Recursion. The Binomial Theorem (Binomial Expansion Formula).	Mathematical Induction. Inductive proofs. Factorials.	Recognize recursive definitions of sequences and sets. Prove elementary properties of recursively defined sets and sequences (Fibonacci and geometric sequences). Recursively construct successive rows of Pascal's triangle. Identify the entries in Pascal's Triangle as Binomial Coefficients. State and apply the Binomial Expansion Formula. Compute individual binomial coefficients using the quotient-of-falling powers formula (n𝑪k) = n(n−1)…(n−k+1)/k!
11	5.1 & 5.2	The rational number system 𝐐. The real number system 𝐑. Fractional powers and roots of real numbers. Rational and irrational numbers. Existence of irrationals.	Divisibility of integers. Unique factorization and the Fundamental Theorem of Arithmetic. Decimals and decimal expansions. Roots and fractional powers of real numbers.	Identify the set 𝐐 of rational numbers as a number system (a field). Identify the set 𝐑 of real numbers as a number system (a field extending 𝐐). Prove the irrationality of √2 and, more generally, of √p for p prime. Prove that fractional powers xm/n of real x>0 are well defined and unique. Informally interpret the convergence of decimal expansions as the completeness of 𝐑. Informally recognize that the universal existence of roots ⁿ√x and fractional powers xm/n of real numbers x>0 relies on the completeness of 𝐑.
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.