MAT3223 - Revision history

Eduardo.duenez: Typo…

2023-03-24T17:00:26Z

Typo…

Eduardo.duenez: Formatting

2023-03-23T19:06:38Z

Formatting

Eduardo.duenez: Styling.

2023-03-23T19:05:22Z

Styling.

Eduardo.duenez: Finished table of topics.

2023-03-23T19:03:44Z

Finished table of topics.

Eduardo.duenez: Up to 2nd midterm (week 10).

2023-03-23T18:44:08Z

Up to 2nd midterm (week 10).

Eduardo.duenez: Up to week 9.

2023-03-23T18:32:37Z

Up to week 9.

Eduardo.duenez: Done til week 8.

2023-03-23T18:26:44Z

Done til week 8.

Eduardo.duenez: Fixing topics for first third of semester.

2023-03-23T18:04:50Z

Fixing topics for first third of semester.

Eduardo.duenez: Topics for first third of semester.

2023-03-23T17:59:30Z

Topics for first third of semester.

Eduardo.duenez: Added textbook information.

2023-03-23T17:06:35Z

Added textbook information.

@@ Line 11: / Line 11: @@
 |1
 ||
-.1 & 2.2
+.1, 2.2
 ||
 Introduction to complex numbers, their operations and geometry.
@@ Line 60: / Line 60: @@
 || <!-- Sections -->
-.3, 4.5 & 4.5
+.3, 4.4 & 4.5
 ||  <!-- Topics -->
 Complex natural logarithms. Multivalued holomorphic functions. Singularities.

@@ 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.
-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"

@@ 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.
-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"

@@ Line 139: / Line 139: @@
 || <!-- Sections -->
-.1–8.4
+.1–8.3
 ||  <!-- Topics -->
-* Introduction to complex numbers and their operations.
+Isolated singularities and Laurent series. The Residue Theorem.
 ||  <!-- 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.
 |-  <!-- START ROW -->
 | <!-- Week# -->
 || <!-- Sections -->
-.5–8.7
+Chapter 9.
 ||  <!-- Topics -->
-* Polar form of complex numbers.
+Calculus of residues.
 ||  <!-- 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.
 |-  <!-- START ROW -->
 | <!-- Week# -->
 || <!-- Sections -->
-.8–9.2
+.1–11.3
 ||  <!-- Topics -->
-* Roots and factors of polynomials. The Remainder Theorem.
+Conformal mappings.
 ||  <!-- SLOs -->
-* State and prove the Remainder Theorem.
+* Preservation of angles and conformal mappings of the plane.
-* Identify roots with linear factors of a polynomial.
+* Conformal mappings yield pairs of conjugate harmonic functions.
-* Factor given simple polynomials into irreducible factors over ℚ, ℝ and ℂ.
+* Dirichlet's Problem on a planar region.
-* State the Fundamental Theorem of Algebra.
+* The Riemann Mapping Theorem.
-* Use the Fundamental Theorem of Algebra to prove that irreducible real polynomials are linear or quadratic.
+* Möbius transformations and their use in solving elementary Dirichlet Problems.
 |-
 |}

@@ Line 50: / Line 50: @@
 .2
 ||  <!-- Topics -->
-Examples of power series of holomorphic functions.
+Examples of power series and their formal manipulation.
 ||  <!-- SLOs -->
 * Review of Taylor coefficients and Taylor series. Radius of convergence.
-* Differentiation of Taylor series.
+<!-- * Differentiation of Taylor series. -->
-* Taylor series of rational functions.
+* Power series of rational functions.
-* The complex exponential, trigonometric and hyperbolic functions and their Taylor series.
+* Power series defining the complex exponential, trigonometric and hyperbolic functions.
 |-  <!-- START ROW -->
 | <!-- Week# -->
@@ Line 66: / Line 66: @@
 ||  <!-- SLOs -->
 * 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.
 * Definition of branch point and branches.
@@ Line 120: / Line 120: @@
 || <!-- Sections -->
-.2 & 4.3
+.1 & 7.2
 ||  <!-- Topics -->
-* Recursion.
+Cauchy's Integral Formula. Taylor series.
-* The Binomial Theorem (Binomial Expansion Formula).
+<!-- Liouville's Theorem. The Fundamental Theorem of Algebra. -->
 ||  <!-- 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.
 |-  <!-- START ROW -->
 | <!-- Week# -->
 || <!-- Sections -->
-.1 & 5.2
+None
 ||  <!-- Topics -->
-* The rational number system 𝐐.
+Review. Second midterm exam.
 |-  <!-- START ROW -->
 | <!-- Week# -->

@@ Line 84: / Line 84: @@
 || <!-- Sections -->
-.5,
+.2 & 5.3
 ||  <!-- Topics -->
-* Arithmetic congruences and basic modular arithmetic.
+Parametric curves. Line integrals.
 ||  <!-- 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.
 |-  <!-- START ROW -->
 | <!-- Week# -->
 || <!-- Sections -->
-.4
+.4 & 5.5
 ||  <!-- Topics -->
-* Modular rings ℤₙ.
+Estimation and convergence of line integrals.
 ||  <!-- 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.
 |-  <!-- START ROW -->
 | <!-- Week# -->

@@ 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.
-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"
-! Week # !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
+! Week # !! Sections !! Topics !! Student Learning Outcomes
 |-
 |1
 ||
-.1 & 1.2
+.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.
 |-  <!-- START ROW -->
 | <!-- Week# -->
 || <!-- Sections -->
-.3 & 1.4
+.1, 3.2, 3.3
 ||  <!-- Topics -->
-* Tautologies and Deductions.
+Topology of the complex plane. Continuous complex functions.
 ||  <!-- 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.
 |-  <!-- START ROW -->
 | <!-- Week# -->
 || <!-- Sections -->
-.5 & 1.6
+.1
 ||  <!-- Topics -->
-* Sets.
+Complex differentiation
 ||  <!-- 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.
 |-  <!-- START ROW -->
 | <!-- Week# -->
 || <!-- Sections -->
-.1
+.2
 ||  <!-- Topics -->
-* Divisibility of integers.
+Power (Taylor) series of holomorphic functions.
 ||  <!-- SLOs -->
-* Recognize the notion of integer divisibility via its formal definition, examples and counterexamples.
+* Taylor coefficients and Taylor series of a holomorphic function.
-* Correctly state and apply the Division Algorithm of integers.
+* Radius of convergence.
-* Prove basic facts pertaining to divisibility and the division algorithm.
+* Differentiation of Taylor series.
 |-  <!-- START ROW -->
 | <!-- Week# -->
 || <!-- Sections -->
-.2 & 2.3
+.3 & 4.4
 ||  <!-- Topics -->
-* Greatest Common Divisor.
+Complex natural logarithms. Multivalued holomorphic functions.
 <!-- * Linear Diophantine equations in two variables. -->
 ||  <!-- 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.
 |-  <!-- START ROW -->
 | <!-- Week# -->
 || <!-- Sections -->
-.5
+None
 ||  <!-- Topics -->
-* Primes.
+Review. First midterm exam.
 ||  <!-- SLOs -->
 |-  <!-- START ROW -->
 | <!-- Week# -->

← Older revision		Revision as of 17:06, 23 March 2023
Line 3:		Line 3:

	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]

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