Lecture Notes
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Lecture 0 (January 6, 2020) Definition of complex numbers. Addition, multiplication, modulus, inverse.
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Lecture 1 (January 8, 2020) Polar coordinates. Multiplication in polar coordinates. de Moivre's formula.
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Lecture 2 (January 10, 2020) n-th roots of a complex number. Triangle inequality.
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Lecture 3 (January 13, 2020) Topological properties: open and closed sets. Connectedness. Accumulation points. Compact sets.
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Lecture 4 (January 15, 2020) Function of a complex variable: limit and continuity.
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Lecture 5 (January 17, 2020) Polynomial and rational functions. Notion of complex differentiability. Cauchy-Riemann equations.
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Lecture 6 (January 22, 2020) Cauchy-Riemann equations continued: proof of sufficiency, polynomials and rational functions are complex differentiable.
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Lecture 7 (January 24, 2020) Laplace equations. Harmonic functions. Properties of derivatives: product rule, chain rule etc.
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Lecture 8 (January 27, 2020) Exponential function. Trigonometric functions. Some trigonometric identities.
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Lecture 9 (January 29, 2020) Logarithm function. Power function.
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Lecture 10 (January 31, 2020) Limits at infinity. The point at infinity - Riemann sphere.
- Lecture 11
(February 7, 2020). Review of the theory of integration.
- Lecture 12
(February 10, 2020). Paths. Line integrals. Basic properties and an
important inequality.
- Lecture 13
(February 12, 2020). Examples. Antiderivatives. Zig-zag paths.
- Lecture 14
(February 14, 2020). Proof of the theorem of antiderivatives.
Closed paths (loops). Simple paths. Contours. Orientation.
- Lecture 15
(February 17, 2020). Cauchy's theorem.
- Lecture 16
(February 19, 2020). Principle of deformation of contours,
Stronger version of Cauchy's theorem. Cauchy's integral formula.
- Lecture 17
(February 21, 2020). Generalization of Cauchy's integral formula.
Once differentiable always differentiable. Name change: holomorphic
functions.
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Lecture 18
(February 24, 2020). Liouville's theorem. Fundamental theorem of algebra.
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Lecture 19
(February 26, 2020). Rational functions. Partial fractions.
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Lecture 20
(February 28, 2020). Maximum modulus principle, Schwarz' lemma,
holomorphic reparametrizations of a disc.
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Lecture 21
(March 6, 2020). Review of sequences and series.
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Lecture 22
Uniform convergence. Power series and radius of convergence (Abel's theorem).
Weierstrass' theorem on uniformly convergent sequence of functions.
Here is the tex file.
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Lecture 23
Power series as functions on open discs. Taylor series of a holomorphic
function.
Here is the tex file.
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Lecture 24
Laurent series near isolated singularity. Three types of
singularities: removable, pole, essential.
Here is the tex file.
and pictures (needed to compile the tex file)
Figure 1 ,
Figure 2 ,
Figure 3
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Lecture 25
Order of a zero. Identity theorem. Poles again. Definition of residue.
Here is the tex file.
and pictures (needed to compile the tex file)
Figure 1 ,
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Lecture 26
Residue at infinity. Non-isolated case. Cauchy's residue theorem.
Meromorphic functions. Meromorphic functions with pole at infinity
are rational.
Here is the tex file.
and pictures (needed to compile the tex file)
Figure 1 ,
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Lecture 27
Applications of residues to real integrals - part 1.
Here is the tex file.
and pictures (needed to compile the tex file)
Figure 1 ,
Figure 2 ,
Figure 3 ,
Figure 4 ,
Figure 5 ,
Figure 6 .
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Lecture 28
Applications of residues part 2. Jordan's lemma. Indented contours.
Here is the tex file.
and pictures:
Figure 1 ,
Figure 2 ,
Figure 3 ,
Figure 4 ,
Figure 5 .
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Lecture 29
A bit more advanced applications: infinite sum/product
expansions of trigonometric functions (Optional).
Here is the tex file.
and pictures:
Figure 1 .
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Lecture 30
Funtions defined by infinite integrals. Laplace transform.
Here is the tex file.
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Lecture 31
Gamma function - part 1. Euler's integral. Difference equation.
Beta function B(p,q).
Here is the tex file
and a figure.
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Lecture 32
Gamma function - part 2. Weierstrass' infinite product
formula. Gamma function vs trigonometric functions.
Here is the tex file.
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Lecture 33
Doubly-periodic functions. Basic properties.
Here is the tex file.
And the figures:
Figure 1
Figure 2
Figure 3
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Lecture 34
Jacobi's theta function. Any doubly-periodic function
can be written in terms of theta function.
Here is the tex file.
And a figure
Figure 1
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Lecture 35
Jacobi's theta function continued. Fay's trisecant identity.
Theta function as infinite series - Jacobi's triple product identity.
Here is the tex file.
And two figures:
Figure 1
Figure 2
Optional reading material
- Appendix A
Bolzano-Weierstrass theorem. Cauchy's criterior for convergence.
Descending chain property. Absolute max/min. Heine-Borel theorem.
Here is the tex file.