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 Ungdergraduate 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 !! Prerequisite Skills !! Student Learning Outcomes
+
! Week !! Sections !! Topics !! Student Learning Outcomes
 
|-                 
 
|-                 
 
|1
 
|1
 
||
 
||
−
1.1 & 1.2
+
2.1, 2.2
 
||
 
||
−
Propositional Logic
+
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).
−
* Recognize propositional formulas built from atoms using connectives.
+
* Complex numbers in Cartesian and polar forms.
−
* Correctly interpret propositional formulas using truth tables.
+
* Complex operations: Elementary algebraic identities and inequalities.
 +
* Geometric meaning of complex arithmetic operations.
 +
* DeMoivre's Formula.
 
|-  <!-- START ROW -->
 
|-  <!-- START ROW -->
 
| <!-- Week# -->
 
| <!-- Week# -->
 
2
 
2
 
|| <!-- Sections -->
 
|| <!-- Sections -->
−
1.3 & 1.4
+
3.1, 3.2, 3.3
 
||  <!-- Topics -->
 
||  <!-- Topics -->
−
* Tautologies and Deductions.
+
Topology of the complex plane. Continuous complex functions.
−
* Quantifiers.
 
−
||  <!-- Prereqs -->
 
−
* Propositional Logic.
 
 
||  <!-- SLOs -->
 
||  <!-- SLOs -->
−
* Establish whether a propositional formula is a tautology.
+
* Essential analysis concepts: sequences, series, limits, convergence, completeness.
−
* State De Morgan's Laws of logic.
+
* Basic topology of the complex plane: open, closed and punctured discs, open and closed sets, neighborhoods.
−
* Recognize conditional tautologies as laws of deduction.
+
* Continuous functions and operations on them.
−
* Express conditionals in disjunctive form.
 
−
* Express the negation of a conditional in conjunctive form.
 
−
* Identify the direct and contrapositive forms of a conditional.
 
−
* Recognize the non-equivalence of a conditional and its converse.
 
−
* Recognize a biconditional as the conjunction of a conditional and its converse.
 
−
* Identify the domain of interpretation of a quantified statement.
 
−
* Correctly interpret quantified statements.
 
−
* Correctly negate quantified statements.
 
 
|-  <!-- START ROW -->
 
|-  <!-- START ROW -->
 
| <!-- Week# -->
 
| <!-- Week# -->
 
3
 
3
 
|| <!-- Sections -->
 
|| <!-- Sections -->
−
1.5 & 1.6
+
4.1
 
||  <!-- Topics -->
 
||  <!-- Topics -->
−
* Sets.
+
Complex differentiation
−
* Set Operations.
 
−
* Introduction to proofs of universal statements in set theory
 
−
* Disproving universal statements via counterexamples.
 
−
||  <!-- Prereqs -->
 
−
* Tautologies and Deductions.
 
−
* Quantifiers.
 
 
||  <!-- SLOs -->
 
||  <!-- SLOs -->
−
* Recognize and interpret set equality and set inclusion.
+
* Definition of complex derivative at a point.
−
* Recognize set operations and state their formal definitions.
+
* Cauchy-Riemann equations.
−
* Recognize formal proofs as processes of logical deduction of conclusions from assumptions.
+
* Examples of differentiable and non-differentiable complex functions.
−
* Prove basic universal statements pertaining to set inclusion and set operations.
+
* Holomorphic functions.
−
* Correctly identify false universal statements in set theory and disprove them with appropriate counterexamples.
 
−
* Correctly use propositional and quantified tautologies as deductive laws.
 
 
|-  <!-- START ROW -->
 
|-  <!-- START ROW -->
 
| <!-- Week# -->
 
| <!-- Week# -->
 
4
 
4
 
|| <!-- Sections -->
 
|| <!-- Sections -->
−
2.1
+
4.2
 
||  <!-- Topics -->
 
||  <!-- Topics -->
−
* Divisibility of integers.
+
Examples of power series and their formal manipulation.
−
* The Division Algorithm.
 
−
||  <!-- Prereqs -->
 
−
* Proofs and Counterexamples.
 
−
* Propositional Logic.
 
−
* Quantifiers.
 
 
||  <!-- SLOs -->
 
||  <!-- SLOs -->
−
* Recognize the notion of integer divisibility via its formal definition, examples and counterexamples.
+
* Review of Taylor coefficients and Taylor series. Radius of convergence.
−
* Correctly state and apply the Division Algorithm of integers.
+
<!-- * Differentiation of Taylor series. -->
−
* Prove basic facts pertaining to divisibility and the division algorithm.
+
* Power series of rational functions.
 +
* Power series defining the complex exponential, trigonometric and hyperbolic functions.
 
|-  <!-- START ROW -->
 
|-  <!-- START ROW -->
 
| <!-- Week# -->
 
| <!-- Week# -->
 
5
 
5
 
|| <!-- Sections -->
 
|| <!-- Sections -->
−
2.2 & 2.3
+
4.3, 4.4 & 4.5
 
||  <!-- Topics -->
 
||  <!-- Topics -->
−
* Greatest Common Divisor.
+
Complex natural logarithms. Multivalued holomorphic functions. Singularities.
−
* Bèzout's Identity: GCD(a,b) = au + bv for some u,v∊ℤ.
 
−
* Coprime integers.
 
−
* The Extended Euclidean Algorithm.
 
 
<!-- * Linear Diophantine equations in two variables. -->
 
<!-- * Linear Diophantine equations in two variables. -->
−
||  <!-- Prereqs -->
 
−
* Divisibility of integers.
 
−
* The Division Algorithm.
 
 
||  <!-- SLOs -->
 
||  <!-- SLOs -->
−
* Compute the GCD of two integers using the Euclidean algorithm.
+
* Definition of the multivalued complex natural logarithm, its principal branch, and other branches.
−
* Express the GCD of two integers as a linear combination thereof using the extended Euclidean algorithm.
+
<!-- * Derivatives of inverse functions. Derivative of the complex natural logarithm. -->
−
<!-- * Solve integer linear equations mu+nv=a. -->
+
* 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 -->
 
|-  <!-- START ROW -->
 
| <!-- Week# -->
 
| <!-- Week# -->
 
6
 
6
 
|| <!-- Sections -->
 
|| <!-- Sections -->
−
2.5
+
None
 
||  <!-- Topics -->
 
||  <!-- Topics -->
−
* Primes.
+
Review. First midterm exam.
−
* Euclid's proof of the infinitude of primes by contradiction.
 
−
* Euclid's Lemma: for p prime, p|ab implies p∣a or p∣b.
 
−
* Unique factorization and the Fundamental Theorem of Arithmetic.
 
−
||  <!-- Prereqs -->
 
−
* Divisibility of integers.
 
−
* The Extended Euclidean Algorithm.
 
−
* Greatest Common Divisor.
 
−
* Coprime integers.
 
 
||  <!-- SLOs -->
 
||  <!-- SLOs -->
−
* Define prime numbers and state their basic properties.
 
−
* Prove the infinitude of primes.
 
−
* Prove Euclid's Lemma using Bèzout's identity.
 
−
* Prove uniqueness of prime factorization using Euclid's Lemma.
 
−
* Characterize divisibility and GCD of integers in terms of their prime factorizations.
 
 
|-  <!-- START ROW -->
 
|-  <!-- START ROW -->
 
| <!-- Week# -->
 
| <!-- Week# -->
 
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.
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