Lecture Notes
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Lecture 00 (August 23, 2023) .
Introduction. Cauchy-Riemann equations. Some historical remarks.
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Lecture 01 (August 25, 2023) .
Laplace equation and harmonic functions. Exponential function.
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Lecture 02 (August 28, 2023) .
Line integrals, Morera's theorem, Cauchy's theorem.
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Lecture 03 (August 30, 2023) .
Principle of contour deformation. Logarithm. Multi-valued
functions.
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Lecture 04 (September 1, 2023) .
Cauchy's integral formula. Liouville's theorem and
the fundamental theorem of algebra.
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Lecture 05 (September 6, 2023) .
Notion of uniform convergence. Weierstrass' theorem.
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Lecture 06 (September 8, 2023) .
Power series. Abel's theorem. Taylor and Laurent
series.
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Lecture 07 (September 11, 2023) .
Meromorphic functions. Behaviour at infinity.
Permanence of relations. Analytic continuation.
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Lecture 08 (September 13, 2023) .
Germs of holomorphic functions. Riemann surface
of a local function. Remarks on the next topic.
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Lecture 09 (September 15, 2023) .
Differential equations. Linear case: regular, regular singular and
irregular points.
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Lecture 10 (September 18, 2023) .
Local behaviour of solutions, near regular singular and irregular
points. Divergence issues.
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Lecture 11 (September 20, 2023) .
Asymptotic (power) series. Sectors. Exponential-integral
series.
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Lecture 12 (September 22, 2023) .
More examples. Difference equations. Summation of a series.
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Lecture 13 (September 25, 2023) .
Functions defined via integrals. Laplace transform and
Watson's lemma.
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Lecture 14 (September 27, 2023) .
Laplace transform along arbitrary rays. Relation with
analytic continuation.
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Lecture 15 (September 29, 2023) .
Laplace transform continued. Jump behaviour, or Stokes'
phenomenon.
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Lecture 16 (October 2, 2023) .
Mittag Leffler's theorem on infinite partial fractions.
Weierstrass' infinite product formula.
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Lecture 17 (October 4, 2023) .
Mittag-Leffler theorem. Euler's gamma function.
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Lecture 18 (October 6, 2023) .
Further properties of gamma function.
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Lecture 19 (October 9, 2023) .
Uniqueness of gamma. Stirling series. (Notes by Liam O'Brien).
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Lecture 20 (October 16, 2023) .
An application of Laplace transforms/Watson's lemma:
n-th order, linear additive difference equations, with constant
coefficients.
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Lecture 21 (October 18, 2023) .
Asymptotic expansions of integrals: Laplace's method.
Revisiting Watson's lemma and examples.
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Lecture 22 (October 20, 2023) .
Laplace formula from Watson's lemma. Example of Stirling
series again. Steepest descent method - example.
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Lecture 23 (October 23, 2023) .
Paths of (steepest) ascent/descent. Local behaviour
near critical points. Another example of steepest descent
technique.
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Lecture 24 (October 25, 2023) .
Airy differential equation. Integral representation
of Airy function. Leading behaviour via steepest
descent paths.
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Lecture 25 (October 27, 2023) .
Asymptotic analysis of Airy function.
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Lecture 26 (October 30, 2023) .
Open mapping and inverse function theorems. Conformal
maps (preservation of angles).
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Lecture 27 (November 1, 2023) .
Maximum modulus principle, area formula, argument principle.
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Lecture 28 (November 3, 2023) .
Rouche's theorem. Casorati-Weierstrass theorem. Schwarz lemma.
Conformal automorphisms of the complex plane and the unit disc.
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Lecture 29 (November 6, 2023) .
Mobius transformations: fundamental properties.
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Lecture 30 (November 8, 2023) .
Mobius transformations cntd. Reflections through lines
and circles.
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Lecture 31 (November 13, 2023) .
Glossary of commonly used conformal equivalences.
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Lecture 32 (November 15, 2023) .
Proofs of Hurwitz', Koebe's and Riemann mapping theorems
(assuming Montel's theorem)
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Lecture 33 (November 17, 2023) .
Dirichlet's boundary value problem. Case of the unit disc.
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Lecture 34 (November 20, 2023) .
Poisson Kernel. Relation with invariant measures on the
circle and hyperbolic metric on the unit disc.
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Lecture 35 (November 27, 2023) .
Boundary behaviour of Riemann maps. Schwarz' reflection
principle.
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Lecture 36 (November 29, 2023) .
Schwarz-Christoffel formula. Examples. Doubly-periodic
functions.
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Lecture 37 (December 4, 2023) .
Doubly-periodic functions. Jacobi's theta function.
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Lecture 38 (December 8, 2023) .
Jacobi's theta function.