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Integral Transforms and Asymptotics Semester 2, 2007 |
2007 Lecture and Practice Class Schedule
This is subject to modification, but please note that- the midsemester test date and venue are fixed;
- practice classes commence in the first week.
- L1 Tue 24 July: Landau symbols, Stirling's approximation.
- P1 Wed 25 July: Landau symbols, convergence of series.
- L2 Thu 26 July: Convergent and divergent asymptotic series. Case studies.
- L3 Fri 27 July: Interchange tricks and repeated integration by parts.
- L4 Tue 31 July: Holomorphic functions, complex power series, complex Taylor's Theorem.
- P2 Wed 1 August: Binomial expansion. Bernoulli numbers.
- L5 Thu 2 August: Contour integration, Cauchy and Laurent Theorems. Residue calculus.
- L6 Fri 3 August: Rational functions integrated over R. Fourier integrals of rational functions.
- L7 Tue 7 August: Rectangular contours and indented contours.
- P3 Wed 8 August: Practical contour integrals.
- L8 Thu 9 August: Branch cuts, Hankel loop contours.
- L9 Fri 10 August: Sector integrals. Gamma function integral properties.
- L10 Tue 14 August: Analytic continuation in general. Continuation of the gamma function.
- P4 Wed 15 August: Rectangular, indented and loop contours
- L11 Thu 16 August: Beta function, duplication and reflection formulae for the gamma function.
- L12 Fri 17 August: The Riemann zeta function.
- L13 Tue 21 August: Survey of transform methods: removal of derivatives, resolution of convolution.
- P5 Wed 22 August: Recognizing and applying the gamma, beta and Riemann zeta functions.
- L14 Thu 23 August: The inversion problem. Riemann--Lebesgue Lemma, Fourier Integral Theorem
- L15 Fri 24 August: Advanced Fourier Transform Theory
- L16 Tue 28 August: Fourier methods in probability; boundary value problems; Green functions.
- P6 Wed 29 August: The (exponential) Fourier transform.
- L17 Thu 30 August: Sine and cosine transforms.
- L18 Fri 31 August: Multiple Fourier transforms; Hankel transforms
- L19 Tue 4 September: Mid-semester Test in Wilson Hall on material up to and including Lecture 16.
- P7 Wed 5 September: Fourier transforms and related transforms.
- L20 Thu 6 September: Review of Laplace transforms; selecting the right transform to use.
- L21 Fri 7 September: Laplace transforms as analytic functions; complex inversion formula.
- L22 Tue 11 September: Heaviside's expansion; Shift Theorems; periodic functions.
- P8 Wed 12 September: Complex variable methods for Laplace transforms.
- L23 Thu 13 September: Image functions with branch cuts.
- L24 Fri 14 September: Laplace transforms in probability and differential equations.
Week 9: Asymptotic Expansions of Ordinary Integrals
- L25 Tue 2 October: Watson's Lemma and applications
- P9 Wed 3 October: Watson's Lemma and repeated integration by parts.
- L26 Thu 4 October: Laplace's method.
- L27 Fri 5 October: The method of stationary phase.
- L28 Tue 9 October: Watson's Lemma for analytic integrands.
- P10 Wed 10 October: Laplace's method and stationary phase.
- L29 Thu 11 October: Introduction to the method of steepest descent.
- L30 Fri 12 October: More on steepest descent.
- L31 Tue 16 October: Mellin transforms as analytic functions; Mellin inversion.
- P11 Wed 17 October: Steepest descent.
- L32 Thu 18 October: Mellin transforms in asymptotics.
- L33 Fri 19 October: Further applications of Mellin transforms.
- L34 Tue 23 October: Differential Equations in the Complex Plane; singular points at infinity
- P12 Wed 24 October: Applications of Mellin transforms in asymptotics.
- L35 Thu 25 October: The WKBJ/LG approximation, dominant balance arguments.
- L36 Fri 26 October: Revision and highlighting of major themes.
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