Webpage of the section LEC0101 of MAT334: Complex Variables at UofT - Fall 2020

Information

Instructor Jean-Baptiste (JB) Campesato
E-Mail campesat [@t) math.toronto.edu
Start the subject with "MAT334:"
Schedule Monday, 10am to 11am
Wednesday, 10am to 11am
Friday, 10am to 11am
Online via Zoom (Credentials sent through Quercus)
Office Hours (via Zoom) Monday, 11am to 12pm
Friday, 11am to 12pm
Course Information On Quercus
Practice problems On Quercus
Unfold all lectures
Hide all lectures

My notes

See below for the slides used in class.
Friday, September 11 - Show
Files Slides
My notes (pp. 1-2)
(there are probably some typos and/or mistakes: just send me an e-mail if you think that something is wrong)
Content

The complex plane ℂ

  • Definition and first properties of ℂ
  • Real and imaginary parts
  • Complex conjugate
Textbook section Section 1.1
Monday, September 14 - Show
Files Slides
My notes (pp. 3-6)
(there are probably some typos and/or mistakes: just send me an e-mail if you think that something is wrong)
Content

The complex plane ℂ (continuation)

  • Polar representation: modulus, principal argument, argument
  • De Moivre's formula
  • Exponential representation
  • square roots and n-th roots
Textbook Section Section 1.1
Wednesday, September 16 - Show
Files Slides
My notes
(there are probably some typos and/or mistakes: just send me an e-mail if you think that something is wrong)
Content

Geometry in the complex plane

  • Complex representation of the dot product and the norm
  • Complex equation of a line
  • Complex equation of a circle
  • Generalized circles
  • Apollonius circles
  • Some transformations: translations, rotations, scaling, inversions
Textbook Section Section 1.2
Friday, September 18 - Show
Files Slides
My notes
More on the Riemann sphere: slides - handout
(there are probably some typos and/or mistakes: just send me an e-mail if you think that something is wrong)
Content

Topology of ℂ

  • Interior, closure, boundary.
  • Open sets, closed sets.
  • (Path)-connectedness.
  • Convex sets, star-shaped sets
  • The extended complex plane: the Riemann sphere
Textbook Section 1.3
Monday, September 21 - Show
Files Slides
My notes
More on the Riemann sphere: slides - handout
A question asked several times in office hour
(there are probably some typos and/or mistakes: just send me an e-mail if you think that something is wrong)
Content

Functions, sequences, series

  • Basic definitions about functions: image, range, preimage, graph, injective, surjective, bijective
  • Limits: definitions, properties
  • Continuity: definition, properties
  • Sequences: limits, properties
  • Series: definition, convergence, absolute convergence, d'Alembert ratio's test
Textbook Section 1.4
Wednesday, September 23 - Show
Files Slides - Handout
My notes (pp. 1-4)
(there are probably some typos and/or mistakes: just send me an e-mail if you think that something is wrong)
Content

Functions, sequences, series (continuation)

  • Series: a few words about series using slides from last Monday

Usual functions

  • The complex exponential function: definition, main properties
  • The complex logarithm function(s): indeterminacies/branches, principal branch
Textbook Section 1.5
Friday, September 25 - Show
Files Slides - Handout
My notes
(there are probably some typos and/or mistakes: just send me an e-mail if you think that something is wrong)
Content

Usual functions (continuation)

  • Complex power functions
  • Complex trigonometric functions
  • Inverse complex trigonometric functions (postpone to next Monday)
  • Complex hyperbolic functions (postpone to next Monday)
Textbook Section 1.5
Monday, September 28 - Show
Files Slides
My notes
(there are probably some typos and/or mistakes: just send me an e-mail if you think that something is wrong)
Content

Usual functions (continuation)

Using slides from last Friday.
  • Inverse complex trigonometric functions
  • Complex hyperbolic functions

Line integrals and Green's theorem

  • Curves: definition, simple curves, closed curves, smooth curves, piecewise-smooth curves, orientation of curves.
  • Jordan curve theorem.
Textbook Section 1.6
Wednesday, September 30 - Show
Files Slides
My notes
(there are probably some typos and/or mistakes: just send me an e-mail if you think that something is wrong)
Content

Line integrals and Green's theorem (continuation)

  • Concatenation/sum of curves, reversed orientation.
  • Complex line integrals: definition, basic properties, reparametrization.
  • Green's theorem in the real plane (from multivariable calculus).
  • Green's theorem in the complex plane.
Textbook Section 1.6
Friday, October 2 - Show
Files Slides (slides 1-6)
My notes (pp. 1-3)
(there are probably some typos and/or mistakes: just send me an e-mail if you think that something is wrong)
Content

Holomorphic/Analytic functions

  • Holomorphic/Analytic/ℂ-differentiable functions: definition
  • Holomorphic/Analytic/ℂ-differentiable functions: the Cauchy-Riemann equations
Textbook Section 2.1
Monday, October 5 - Show
Files Slides
My notes
(there are probably some typos and/or mistakes: just send me an e-mail if you think that something is wrong)
Content

Holomorphic/Analytic functions (continuation)

  • Holomorphic/Analytic/ℂ-differentiable functions: examples.
  • Harmonic functions and conjugate harmonic functions: definition.
  • Theorem: the real and imaginary parts of a holomorphic function are conjugate harmonic functions.
  • Theorem: if two functions are harmonic conjugate to a same harmonic function, then they differ by a constant.
  • Counter-example: without assumpion on the domain, a harmonic function may not be the real part of a holomorphic function.
  • Theorem: on a star-shaped (or more generally simply connected) domain a harmonic function is the real part of a holomorphic function.
Textbook Section 2.1
Wednesday, October 7 - Show
Files Slides (pp. 1-11)
My notes (pp. 1-2)
(there are probably some typos and/or mistakes: just send me an e-mail if you think that something is wrong)
Content

Holomorphic/Analytic functions (continuation)

Using slides from last lecture.
  • Theorem: if the range of a holomorphic function defined on a domain lies on a horizontal line, or on a vertical line, or on a circle, then the function is constant.

Power series

  • Definition: power series centered at z0.
  • Radius of convergence (definition, existence, uniqueness, characterizations).
  • D'Alembert ratio test and the root test for power series.
Textbook Section 2.2
Friday, October 9 - Show
Files Slides
My notes
(there are probably some typos and/or mistakes: just send me an e-mail if you think that something is wrong)
Content

Power series (continuation)

  • Sum of two power series.
  • Product of two power series.
  • Power series are holomorphic on their open disk of convergence.
  • Connection between the coefficients of a power series and its successive derivatives.
Textbook Section 2.2
Wednesday, October 14 - Show
Files Slides
My notes
(there are probably some typos and/or mistakes: just send me an e-mail if you think that something is wrong)
Content

Cauchy's integral formula

  • Simple connectedness
  • Cauchy's integral theorem.
  • Consequences on a simply connected domain: path-independence of the line integral of a holomorphic function, existence of a primitive/antiderivative.
Textbook Section 2.3
Friday, October 16 - Show
Files Slides - Handout (slides 1-5)
My notes (§1)
(there are probably some typos and/or mistakes: just send me an e-mail if you think that something is wrong)
Content

Cauchy's integral formula (continuation)

Using slides from last Wednesday.
  • Cauchy's integral formula: statement, proof, example of usage.

Consequences of Cauchy's integral formula

Holomorphic functions are analytic

  • Theorem: a holomorphic function can be locally expressed as a power series (locally = in some neighborhood of each point of its domain).
  • Characterization: a function defined on a open subset of the complex plane is holomorphic if and only if it can be locally expressed as a power series (i.e. in some neighborhood of each point of its domain).
  • Corollary: a holomorphic function is infinitely many times complex differentiable.
  • Counter-examples for real differentiability: a real function may be real differentiable but its (real) derivative is not even continuous, or a real function may be infinitely many times differentiable but not analytic.
Textbook Section 2.4
Monday, October 19 - Show
Files Slides - Handout (slides 6-10)
My notes (pp. 2-4)
(there are probably some typos and/or mistakes: just send me an e-mail if you think that something is wrong)
Content

Consequences of Cauchy's integral formula (continuation)

Continuation of analytic functions

Order of a zero

Textbook Section 2.4
Wednesday, October 21 - Show
Files Slides - Handout (slides 11-18)
My notes (pp. 4-7)
(there are probably some typos and/or mistakes: just send me an e-mail if you think that something is wrong)
Content

Consequences of Cauchy's integral formula (continuation)

Morera's theorem

The characterizations of holomorphicity/analyticity we encountered this term!

Liouville's theorem

Application: the Fundamental Theorem of Algebra.

Analytic logarithm

Elementary proof of the multiplication theorem for power series

Textbook Section 2.4
Friday, October 23 - Show
Files Slides
My notes
(there are probably some typos and/or mistakes: just send me an e-mail if you think that something is wrong)
Content

Isolated singularities

  • Removable singularities: definition and characterizations (Riemann's removable singularity theorem).
  • Poles: definition, characterizations, order of a pole.
  • Essential singularities: definition, Great Picard's theorem (not part of MAT334)
Textbook Section 2.5
Monday, October 26 - Show
Files Isolated singularities:
Slides
My notes
Laurent series:
Slides - Handout
My notes
(there are probably some typos and/or mistakes: just send me an e-mail if you think that something is wrong)
Content

Isolated singularities

  • Removable singularities: definition and characterizations (Riemann's removable singularity theorem).
  • Poles: definition, characterizations, order of a pole.
  • Essential singularities: definition, Great Picard's theorem (not part of MAT334)

Laurent series and residues

  • Laurent's theorem
Textbook Section 2.5
Wednesday, October 28 - Show
Files Slides - Handout
My notes
(there are probably some typos and/or mistakes: just send me an e-mail if you think that something is wrong)
Content

Laurent series and residues

  • Laurent's theorem (proof)
  • Residues: definition, characterization, how to compute them
  • Residue at infinity
Textbook Section 2.5
Friday, October 30 - Show
Files Slides - Handout
My notes
(there are probably some typos and/or mistakes: just send me an e-mail if you think that something is wrong)
Content

Cauchy's residue theorem

  • Statement and proof
Textbook Section 2.6
Monday, November 2 - Show
Files Slides - Handout
My notes
(there are probably some typos and/or mistakes: just send me an e-mail if you think that something is wrong)
Content

Cauchy's residue theorem (continuation)

  • Examples
Textbook Section 2.6
Wednesday, November 4 - Show
Files Slides (slides 1-4)
My notes (pp. 1-2)
(there are probably some typos and/or mistakes: just send me an e-mail if you think that something is wrong)
Content

Zeroes of analytic functions

  • Isolated zeroes
Textbook Section 3.1
Friday, November 6 - Show
Files Slides
My notes
(there are probably some typos and/or mistakes: just send me an e-mail if you think that something is wrong)
Content

Zeroes of analytic functions (continuation)

  • Isolated zeroes
Textbook Section 3.1
Monday, November 16 - Show
Files Slides
My notes
(there are probably some typos and/or mistakes: just send me an e-mail if you think that something is wrong)
Content

Zeroes of analytic functions (continuation)

  • Logarithmic residues
  • The argument principle
  • Rouché's theorem
  • Another proof of the Fundamental Theorem of Algebra
Textbook Section 3.1
Wednesday, November 18 - Show
Files Slides (slides 1-5)
My notes (pp. 1-2)
(there are probably some typos and/or mistakes: just send me an e-mail if you think that something is wrong)
Content

The Maximum Modulus Principle and the Mean Value Property

  • The open mapping theorem.
  • The maximum modulus principle and corollaries.
  • Schwarz’s lemma
Textbook Section 3.2
Friday, November 20 - Show
Files Slides
My notes
(there are probably some typos and/or mistakes: just send me an e-mail if you think that something is wrong)
Content

The Maximum Modulus Principle and the Mean Value Property (continuation)

  • The Maximum Modulus Principle and the boundary of the domain.
  • The mean value property for holomorphic functions.
  • The mean value property for harmonic functions.
Textbook Section 3.2
Monday, November 23 - Show
Files Slides (1-5)
My notes (pp. 1-2)
(there are probably some typos and/or mistakes: just send me an e-mail if you think that something is wrong)
Content

Linear fractional transformations

  • Definition
  • Basic properties (the composition of two Möbius transformations is a Möbius transformation, a Möbius transformation is bijective and its inverse is a Möbius transformation too)
  • Matrix representation
Textbook Section 3.3
Wednesday, November 25 - Show
Files Slides (6-9)
My notes (pp. 2-3)
(there are probably some typos and/or mistakes: just send me an e-mail if you think that something is wrong)
Content

Linear fractional transformations (continuation)

  • Fixed points
  • Triples
  • Lines and Circles
Textbook Section 3.3
Friday, November 27 - Show
Files Linear fractional transformations:
Slides
My notes
(there are probably some typos and/or mistakes: just send me an e-mail if you think that something is wrong)
Content

Linear fractional transformations (continuation)

  • Möbius transformations and the cross ratio
  • Automorphisms of the unit disk
Textbook Sections 3.3
Monday, November 30 - Show
Files Slides
My notes
(there are probably some typos and/or mistakes: just send me an e-mail if you think that something is wrong)
Content

Conformal mappings

  • Definition, geometric interpretation, examples.
  • Connection between holomorphicity and conformality.
Textbook Section 3.4
Wednesday, December 2 - Show
Files Conformal mappings:
Slides
My notes
The Riemann mapping theorem:
Slides
My notes
(there are probably some typos and/or mistakes: just send me an e-mail if you think that something is wrong)
Content

Conformal mappings (continuation)

  • Connection between holomorphicity and conformality.
  • A holomorphic function is conformal at z0 if and only if f'(z0)
  • A global result: if f is injective and holomorphic then f' never vanishes.
    As a corollary, an injective holomorphic function is conformal.

The Riemann mapping theorem

  • The Riemann mapping theorem: a simply connected subset of ℂ which is not ℂ is conformally equivalent to the open unit disk.
  • By Liouville's theorem: ℂ is NOT conformally equivalent to the open unit disk.
Textbook Section 3.4, 3.5
Friday, December 4 - Show
Files Slides
My notes
(there are probably some typos and/or mistakes: just send me an e-mail if you think that something is wrong)
Content

The Riemann mapping theorem (continuation)

Textbook Section 3.5
Monday, December 7 - Show
Files Slides
My notes
(there are probably some typos and/or mistakes: just send me an e-mail if you think that something is wrong)
Content

The Schwarz-Christoffel formula

Textbook Section 3.5
Wednesday, December 9 and Thursday, December 10 (Thanksgiving Monday make up) - Show
Content

Reviews

  • Problem 7 from Final Exam 2018F
  • Questions 5 and 14 from Section 3.1 of the textbook
  • Problem 3 From Test 2 2020S + compute residues
  • Problem 6 from Final Exam 2018F
  • Problem 3 from Test 2
  • Problem 6 from Final Exam 2020S
  • Computing an improper integral using the residue theorem