MAT3223

From Department of Mathematics at UTSA
Revision as of 11:59, 23 March 2023 by Eduardo.duenez (talk | contribs) (Topics for first third of semester.)
Jump to navigation Jump to search

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 & 4.4

Complex natural logarithms. Multivalued holomorphic functions.

  • 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. Examples via complex logarithms, inverse trigonometric/hyperbolic functions, and complex powers.

6

None

Review. First midterm exam.

7

3.1–3.3

  • 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.