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ADVANCED MATHEMATICAL METHODS

J. S. B. Gajjar
j.gajjar@manchester.ac.uk
http://www.maths.manchester.ac.uk/ ~gajjar

August 23, 2012

Contents
1 Introduction
 1.1 Scope of course
 1.2 Important definitions and preliminaries
2 Basic complex analysis
 2.1 Singularities of complex functions
 2.2 Cauchy’s residue and other important theorems
 2.3 Use of Cauchy’s residue theorem in evaluation of integrals
 2.4 Jordan’s Lemma
 2.5 Plemlj formulae
3 Frobenius method
 3.1 Introduction
 3.2 Classification of singularities
 3.3 Properties near ordinary and regular singular points
 3.4 Frobenius solution for 2nd order odes
 3.5 Roots differ by an integer, γ2 - γ1 = N
 3.6 Roots differ by an integer γ2 - γ1 = N > 0
4 Behaviour near irregular singular points
 4.1 Method of dominant balance
 4.2 Method of Dominant balance
5 Some special functions
 5.1 Airy Functions
 5.2 Stokes’s Phenomenon
 5.3 Linear relations between Airy functions
 5.4 Integral represensations of Airy functions
 5.5 Properties of parabolic cylinder functions
6 Properties of the Gamma function
 6.1 Definition of the Gamma Function due to Weierstrass (1856)
 6.2 Euler’s (1729) definition of the Gamma function
 6.3 Integral representation of the Gamma function
 6.4 Mittag-Leffler (1880) expansions and infinite products
 6.5 Hankel’s (1864) loop integral for Γ(z).
 6.6 Stirling’s formula for Γ(z) for large z.
7 Matched expansions, Boundary Layer Theory, WKB method.
 7.1 Boundary layer theory - regular and singular perturbation problems.
 7.2 Uniform approximations
 7.3 More on matching and intermediate variables
 7.4 Interior boundary layers
 7.5 The LG approximation, WKBJ Method
Bibliography
8 Introduction to generalised Functions
 8.1 Introduction
 8.2 Derivatives of generalised functions
 8.3 Application to singular integrals
9 Integral Transforms
 9.1 Fourier Transform
 9.2 Laplace Transform
 9.3 Mellin Transform
 9.4 Riemann-Lebesgue Lemma, and analytic continuation of Mellin transforms.
 9.5 Analytic continuation of Mellin transforms
10 Asymptotic expansion of integrals
 10.1 Asymptotic expansion of integrals using the Mellin transform technique
 10.2 Laplace’s Method
 10.3 Method of stationary phase
 10.4 Method of steepest descent
11 Introduction to Wiener-Hopf method
 11.1 Conformal mapping
 11.2 Riemann-Hilbert problems and the Wiener-Hopf method
Bibliography