Difference between revisions of "MAT3223"

From Department of Mathematics at UTSA
Jump to navigation Jump to search
(Topics for first third of semester.)
(Typo…)
 
(7 intermediate revisions by the same user not shown)
Line 4: Line 4:
 
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.
 
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. [https://link.springer.com/book/10.1007/978-1-4471-0027-0]
+
Textbook: John M. Howie, ''Complex Analysis,'' Springer Undergraduate Mathematics Series, Springer-Verlag London (2003). ISBN: 978-1-4471-0027-0. [https://link.springer.com/book/10.1007/978-1-4471-0027-0]
  
 
{| class="wikitable sortable"
 
{| class="wikitable sortable"
−
! Week # !! Sections !! Topics !! Student Learning Outcomes
+
! Week !! Sections !! Topics !! Student Learning Outcomes
 
|-                 
 
|-                 
 
|1
 
|1
 
||
 
||
−
2.1 & 2.2
+
2.1, 2.2
 
||
 
||
 
Introduction to complex numbers, their operations and geometry.
 
Introduction to complex numbers, their operations and geometry.
Line 50: Line 50:
 
4.2
 
4.2
 
||  <!-- Topics -->
 
||  <!-- Topics -->
−
Power (Taylor) series of holomorphic functions.
+
Examples of power series and their formal manipulation.
 
||  <!-- SLOs -->
 
||  <!-- SLOs -->
−
* Taylor coefficients and Taylor series of a holomorphic function.
+
* Review of Taylor coefficients and Taylor series. Radius of convergence.
−
* Radius of convergence.
+
<!-- * Differentiation of Taylor series. -->
−
* Differentiation of Taylor series.
+
* Power series of rational functions.
−
* Taylor series of rational functions.
+
* Power series defining the complex exponential, trigonometric and hyperbolic functions.
−
* The complex exponential, trigonometric and hyperbolic functions and their Taylor series.
 
 
|-  <!-- START ROW -->
 
|-  <!-- START ROW -->
 
| <!-- Week# -->
 
| <!-- Week# -->
 
5
 
5
 
|| <!-- Sections -->
 
|| <!-- Sections -->
−
4.3 & 4.4
+
4.3, 4.4 & 4.5
 
||  <!-- Topics -->
 
||  <!-- Topics -->
−
Complex natural logarithms. Multivalued holomorphic functions.
+
Complex natural logarithms. Multivalued holomorphic functions. Singularities.
 
<!-- * Linear Diophantine equations in two variables. -->
 
<!-- * Linear Diophantine equations in two variables. -->
 
||  <!-- SLOs -->
 
||  <!-- SLOs -->
 
* Definition of the multivalued complex natural logarithm, its principal branch, and other branches.
 
* Definition of the multivalued complex natural logarithm, its principal branch, and other branches.
−
* Derivatives of inverse functions. Derivative of the complex natural logarithm.
+
<!-- * Derivatives of inverse functions. Derivative of the complex natural logarithm. -->
 
* Complex powers via logarithms.
 
* Complex powers via logarithms.
−
* Definition of branch point and branches. Examples via complex logarithms, inverse trigonometric/hyperbolic functions, and complex powers.
+
* 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).
 
|-  <!-- START ROW -->
 
|-  <!-- START ROW -->
 
| <!-- Week# -->
 
| <!-- Week# -->
Line 82: Line 83:
 
7
 
7
 
|| <!-- Sections -->
 
|| <!-- Sections -->
−
3.1–3.3
+
5.2 & 5.3  
 
||  <!-- Topics -->
 
||  <!-- Topics -->
−
* Arithmetic congruences and basic modular arithmetic.
+
Parametric curves. Line integrals.
−
* Tests of divisibility.
 
−
||  <!-- Prereqs -->
 
−
* Divisibility of integers.
 
−
* The Division Algorithm.
 
 
||  <!-- SLOs -->
 
||  <!-- SLOs -->
−
* Use arithmetic congruences to interpret the remainder of integer division.
+
<!-- * Compact subsets of the complex plane. -->
−
* Use congruences to compute remainders of divisions where the quotient is large or irrelevant.
+
<!-- * The Heine-Borel Theorem. -->
−
* Prove basic divisibility criteria by 2, 3, 5, 9 and 11 for number in base 10, using modular arithmetic.
+
* Parametric representation of piecewise smooth curves.
 +
* Arc-length. Rectifiable curves.
 +
* Line integrals: Definition, examples, and elementary properties.
 +
* Line integrals of holomorphic functions. Fundamental Theorem.
 
|-  <!-- START ROW -->
 
|-  <!-- START ROW -->
 
| <!-- Week# -->
 
| <!-- Week# -->
 
8
 
8
 
|| <!-- Sections -->
 
|| <!-- Sections -->
−
3.4
+
5.4 & 5.5
 
||  <!-- Topics -->
 
||  <!-- Topics -->
−
* Modular rings ℤₙ.
+
Estimation and convergence of line integrals.
−
* Modular fields ℤₚ.
 
−
* Fermat's Little Theorem.
 
−
||  <!-- Prereqs -->
 
−
* Primes.
 
−
* Arithmetic congruences and basic modular arithmetic.
 
 
||  <!-- SLOs -->
 
||  <!-- SLOs -->
−
* Recognize the modular rings ℤₙ as number systems.
+
* Majorization of path integrals by arclength and bound on magnitude of integrand.
−
* Evaluate sums, differences, negations and products in ℤₙ.
+
* Antiderivatives of complex functions with path-independent line integrals.
−
* Identify invertible and non-invertible elements of ℤₙ.
+
* Uniform and non-uniform convergence of sequences and series of complex functions.
−
* Find the inverse (when defined) of a given element of ℤₙ.
+
* Continuous uniform limits of continuous sequences and series, and their integrals.
−
* 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.
 
 
|-  <!-- START ROW -->
 
|-  <!-- START ROW -->
 
| <!-- Week# -->
 
| <!-- Week# -->
 
9
 
9
 
|| <!-- Sections -->
 
|| <!-- Sections -->
−
4.1
+
6.1, 6.2, 6.3
 
||  <!-- Topics -->
 
||  <!-- Topics -->
−
* Mathematical Induction.
+
Cauchy's Theorem and its basic consequences.
−
* Inductive proofs.
 
−
||  <!-- Prereqs -->
 
−
* Basic proofs.
 
−
* Tautologies and Deductions.
 
−
* Quantifiers.
 
−
* Divisibility of integers.
 
 
||  <!-- SLOs -->
 
||  <!-- SLOs -->
−
* State the Principle of Mathematical Induction (PMI).
+
* Statement of Cauchy's Theorem.
−
* Prove elementary algebraic and arithmetic statements by induction.
+
* Proof of Cauchy's Theorem.  
−
* Prove elementary algebraic and arithmetic statements by strong induction.
+
* The Deformation Theorem.
 
|-  <!-- START ROW -->
 
|-  <!-- START ROW -->
 
| <!-- Week# -->
 
| <!-- Week# -->
 
10
 
10
 
|| <!-- Sections -->
 
|| <!-- Sections -->
−
4.2 & 4.3
+
7.1 & 7.2
 
||  <!-- Topics -->
 
||  <!-- Topics -->
−
* Recursion.
+
Cauchy's Integral Formula. Taylor series.
−
* The Binomial Theorem (Binomial Expansion Formula).
+
<!-- Liouville's Theorem. The Fundamental Theorem of Algebra. -->
−
||  <!-- Prereqs -->
 
−
* Mathematical Induction.
 
−
* Inductive proofs.
 
−
* Factorials.
 
 
||  <!-- SLOs -->
 
||  <!-- SLOs -->
−
* Recognize recursive definitions of sequences and sets.
+
* Statement and proof of Cauchy's Integral Formula.
−
* Prove elementary properties of recursively defined sets and sequences (Fibonacci and geometric sequences).
+
* Existence, uniqueness, and general theory of Taylor series of holomorphic functions.
−
* Recursively construct successive rows of Pascal's triangle.
+
* Rigorous definition of and proof that complex logarithms are holomorphic.
−
* 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!
 
 
|-  <!-- START ROW -->
 
|-  <!-- START ROW -->
 
| <!-- Week# -->
 
| <!-- Week# -->
 
11
 
11
 
|| <!-- Sections -->
 
|| <!-- Sections -->
−
5.1 & 5.2
+
None
 
||  <!-- Topics -->
 
||  <!-- Topics -->
−
* The rational number system 𝐐.
+
Review. Second midterm exam.
−
* The real number system 𝐑.
 
−
* Fractional powers and roots of real numbers.
 
−
* Rational and irrational numbers. Existence of irrationals.
 
−
||  <!-- Prereqs -->
 
−
* Divisibility of integers.
 
−
* Unique factorization and the Fundamental Theorem of Arithmetic.
 
−
* Decimals and decimal expansions.
 
−
* Roots and fractional powers of real numbers.
 
−
||  <!-- SLOs -->
 
−
* 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 x<sup>m/n</sup> 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 x<sup>m/n</sup> of real numbers x>0 relies on the completeness of 𝐑.
 
 
|-  <!-- START ROW -->
 
|-  <!-- START ROW -->
 
| <!-- Week# -->
 
| <!-- Week# -->
 
12
 
12
 
|| <!-- Sections -->
 
|| <!-- Sections -->
−
8.1–8.4
+
8.1–8.3
 
||  <!-- Topics -->
 
||  <!-- Topics -->
−
* Introduction to complex numbers and their operations.
+
Isolated singularities and Laurent series. The Residue Theorem.
−
* The complex number system 𝐂.
 
−
* The complex plane.
 
−
||  <!-- Prereqs -->
 
−
* The real number system 𝐑.
 
−
* Fractional powers and roots of real numbers.
 
 
||  <!-- SLOs -->
 
||  <!-- SLOs -->
−
* Represent complex numbers algebraically in Cartesian form.
+
* Definition of Laurent series about an isolated singularity. Examples.
−
* Represent complex numbers geometrically as points on a plane.
+
* Types of isolated singularities: Removable, polar, essential. The Cassorati-Weierstrass Theorem.
−
* Carry out arithmetic operations with complex numbers.
+
* Statement and proof of the Residue Theorem.
−
* Interpret the geometric meaning of addition, subtraction and complex conjugation.
+
* Elementary techniques to evaluate residues.
−
* Identify the set 𝐂 of complex numbers as a field extending the real number system 𝐑.
 
 
|-  <!-- START ROW -->
 
|-  <!-- START ROW -->
 
| <!-- Week# -->
 
| <!-- Week# -->
 
13
 
13
 
|| <!-- Sections -->
 
|| <!-- Sections -->
−
8.5–8.7
+
Chapter 9.
 
||  <!-- Topics -->
 
||  <!-- Topics -->
−
* Polar form of complex numbers.
+
Calculus of residues.
−
* Geometric meaning of complex multiplication and division.
 
−
* Powers and roots of complex numbers. De Moivre’s Theorem.
 
−
||  <!-- Prereqs -->
 
−
* The complex number system 𝐂.
 
−
* The complex plane.
 
−
* Roots and fractional powers of real numbers.
 
 
||  <!-- SLOs -->
 
||  <!-- SLOs -->
−
* Represent complex numbers in polar form.
+
* Evaluation of integrals of real analytic functions using residues.
−
* Algebraically relate the Cartesian and polar forms of a complex number.
+
* Evaluation of series of real analytic functions using residues.
−
* Use the identities cis(𝜃+ɸ) = cis𝜃∙cisɸ and (cis𝜃)<sup>n</sup> = 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.
 
 
|-  <!-- START ROW -->
 
|-  <!-- START ROW -->
 
| <!-- Week# -->
 
| <!-- Week# -->
 
14
 
14
 
|| <!-- Sections -->
 
|| <!-- Sections -->
−
8.8–9.2
+
11.1–11.3
 +
||  <!-- Topics -->
 +
Conformal mappings.
 +
||  <!-- SLOs -->
 +
* 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.
 +
|-  <!-- START ROW -->
 +
| <!-- Week# -->
 +
15
 +
|| <!-- Sections -->
 +
Chapter 10. (At instructor's discretion, week 15 may be used to wrap-up and review instead.)
 
||  <!-- Topics -->
 
||  <!-- Topics -->
−
* Roots and factors of polynomials. The Remainder Theorem.
+
Complex integration and geometric properties of holomorphic functions  
−
* Real and complex roots.
 
−
* The Fundamental Theorem of Algebra.
 
−
|| <!-- Prereqs -->
 
−
* The complex number system 𝐂.
 
−
* Powers and roots of complex numbers. De Moivre’s Theorem.
 
−
* Polynomials: arithmetic operations, long division, and factorizations.
 
 
||  <!-- SLOs -->
 
||  <!-- SLOs -->
−
* State and prove the Remainder Theorem.
+
* Rouché's Theorem.
−
* Identify roots with linear factors of a polynomial.
+
* The Open Mapping Theorem.
−
* Factor given simple polynomials into irreducible factors over ℚ, ℝ and ℂ.
+
* Winding numbers.
−
* State the Fundamental Theorem of Algebra.
 
−
* Use the Fundamental Theorem of Algebra to prove that irreducible real polynomials are linear or quadratic.
 
 
|-
 
|-
 
|}
 
|}

Latest revision as of 11:00, 24 March 2023

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, 4.4 & 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.
Retrieved from "https://mathresearch.utsa.edu/wiki/index.php?title=MAT3223&oldid=4924"