The complex plane ℂ

• Definition and first properties of ℂ
• Real and imaginary parts
• Complex conjugate

The complex plane ℂ (continuation)

• Polar representation: modulus, principal argument, argument
• De Moivre's formula
• Exponential representation
• square roots and n-th roots

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

Topology of ℂ

• Interior, closure, boundary.
• Open sets, closed sets.
• (Path)-connectedness.
• Convex sets, star-shaped sets
• The extended complex plane: the Riemann sphere

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

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

Usual functions (continuation)

• Complex power functions
• Complex trigonometric functions
• Inverse complex trigonometric functions (postpone to next Monday)
• Complex hyperbolic functions (postpone to next Monday)

Usual functions (continuation)

• 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.

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.

Holomorphic/Analytic functions

• Holomorphic/Analytic/ℂ-differentiable functions: definition
• Holomorphic/Analytic/ℂ-differentiable functions: the Cauchy-Riemann equations

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.

Holomorphic/Analytic functions (continuation)

• 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.

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.

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.

Cauchy's integral formula (continuation)

• 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.

Consequences of Cauchy's integral formula (continuation)

Continuation of analytic functions

Order of a zero

Consequences of Cauchy's integral formula (continuation)

Morera's theorem

The characterizations of holomorphicity/analyticity we encountered this term!

Liouville's theorem

Analytic logarithm

Elementary proof of the multiplication theorem for power series

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)

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

Laurent series and residues

• Laurent's theorem (proof)
• Residues: definition, characterization, how to compute them
• Residue at infinity

Cauchy's residue theorem

• Statement and proof

Cauchy's residue theorem (continuation)

• Examples

Zeroes of analytic functions

• Isolated zeroes

Zeroes of analytic functions (continuation)

• Isolated zeroes

Zeroes of analytic functions (continuation)

• Logarithmic residues
• The argument principle
• Rouché's theorem
• Another proof of the Fundamental Theorem of Algebra

The Maximum Modulus Principle and the Mean Value Property

• The open mapping theorem.
• The maximum modulus principle and corollaries.
• Schwarz’s lemma

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.

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

Linear fractional transformations (continuation)

• Fixed points
• Triples
• Lines and Circles

Linear fractional transformations (continuation)

• Möbius transformations and the cross ratio
• Automorphisms of the unit disk

Conformal mappings

• Definition, geometric interpretation, examples.
• Connection between holomorphicity and conformality.

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.

The Riemann mapping theorem (continuation)

The Schwarz-Christoffel formula

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