Difference between revisions of "MAT3223"

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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]
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 +
{| class="wikitable sortable"
 +
! 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.
 +
|-  <!-- START ROW -->
 +
| <!-- Week# -->
 +
2
 +
|| <!-- Sections -->
 +
3.1, 3.2, 3.3
 +
||  <!-- Topics -->
 +
Topology of the complex plane. Continuous complex functions.
 +
||  <!-- SLOs -->
 +
* 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.
 +
|-  <!-- START ROW -->
 +
| <!-- Week# -->
 +
3
 +
|| <!-- Sections -->
 +
4.1
 +
||  <!-- Topics -->
 +
Complex differentiation
 +
||  <!-- SLOs -->
 +
* Definition of complex derivative at a point.
 +
* Cauchy-Riemann equations.
 +
* Examples of differentiable and non-differentiable complex functions.
 +
* Holomorphic functions.
 +
|-  <!-- START ROW -->
 +
| <!-- Week# -->
 +
4
 +
|| <!-- Sections -->
 +
4.2
 +
||  <!-- Topics -->
 +
Examples of power series and their formal manipulation.
 +
||  <!-- SLOs -->
 +
* Review of Taylor coefficients and Taylor series. Radius of convergence.
 +
<!-- * Differentiation of Taylor series. -->
 +
* Power series of rational functions.
 +
* Power series defining the complex exponential, trigonometric and hyperbolic functions.
 +
|-  <!-- START ROW -->
 +
| <!-- Week# -->
 +
5
 +
|| <!-- Sections -->
 +
4.3, 4.4 & 4.5
 +
||  <!-- Topics -->
 +
Complex natural logarithms. Multivalued holomorphic functions. Singularities.
 +
<!-- * Linear Diophantine equations in two variables. -->
 +
||  <!-- SLOs -->
 +
* 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).
 +
|-  <!-- START ROW -->
 +
| <!-- Week# -->
 +
6
 +
|| <!-- Sections -->
 +
None
 +
||  <!-- Topics -->
 +
Review. First midterm exam.
 +
||  <!-- SLOs -->
 +
|-  <!-- START ROW -->
 +
| <!-- Week# -->
 +
7
 +
|| <!-- Sections -->
 +
5.2 & 5.3
 +
||  <!-- Topics -->
 +
Parametric curves. Line integrals.
 +
||  <!-- SLOs -->
 +
<!-- * Compact subsets of the complex plane. -->
 +
<!-- * The Heine-Borel Theorem. -->
 +
* 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 -->
 +
| <!-- Week# -->
 +
8
 +
|| <!-- Sections -->
 +
5.4 & 5.5
 +
||  <!-- Topics -->
 +
Estimation and convergence of line integrals.
 +
||  <!-- SLOs -->
 +
* 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.
 +
|-  <!-- START ROW -->
 +
| <!-- Week# -->
 +
9
 +
|| <!-- Sections -->
 +
6.1, 6.2, 6.3
 +
||  <!-- Topics -->
 +
Cauchy's Theorem and its basic consequences.
 +
||  <!-- SLOs -->
 +
* Statement of Cauchy's Theorem.
 +
* Proof of Cauchy's Theorem.
 +
* The Deformation Theorem.
 +
|-  <!-- START ROW -->
 +
| <!-- Week# -->
 +
10
 +
|| <!-- Sections -->
 +
7.1 & 7.2
 +
||  <!-- Topics -->
 +
Cauchy's Integral Formula. Taylor series.
 +
<!-- Liouville's Theorem. The Fundamental Theorem of Algebra. -->
 +
||  <!-- SLOs -->
 +
* 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.
 +
|-  <!-- START ROW -->
 +
| <!-- Week# -->
 +
11
 +
|| <!-- Sections -->
 +
None
 +
||  <!-- Topics -->
 +
Review. Second midterm exam.
 +
|-  <!-- START ROW -->
 +
| <!-- Week# -->
 +
12
 +
|| <!-- Sections -->
 +
8.1–8.3
 +
||  <!-- Topics -->
 +
Isolated singularities and Laurent series. The Residue Theorem.
 +
||  <!-- SLOs -->
 +
* 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.
 +
|-  <!-- START ROW -->
 +
| <!-- Week# -->
 +
13
 +
|| <!-- Sections -->
 +
Chapter 9.
 +
||  <!-- Topics -->
 +
Calculus of residues.
 +
||  <!-- SLOs -->
 +
* Evaluation of integrals of real analytic functions using residues.
 +
* Evaluation of series of real analytic functions using residues.
 +
|-  <!-- START ROW -->
 +
| <!-- Week# -->
 +
14
 +
|| <!-- Sections -->
 +
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 -->
 +
Complex integration and geometric properties of holomorphic functions 
 +
||  <!-- SLOs -->
 +
* Rouché's Theorem.
 +
* The Open Mapping Theorem.
 +
* Winding numbers.
 +
|-
 +
|}

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.
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