Topics in Mathematical Methods
Lecture Notes
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\[\Gamma (z)\,\Gamma (1-z) \;=\; \frac {\pi }{\sin \pi z}, \qquad 0 < \operatorname {Re} z < 1\]
TABLE of CONTENTS
1 Improper Integrals
1.1 Improper Integrals of the First Kind
1.2 Improper Integrals of The Second Kind
1.3 Convergence and Divergence of Improper Integrals
1.4 Integrals of the Second Kind
1.5 Integrals of Mixed Type
1.6 Practice Problems
2 Special Functions
2.1 The Gamma Function
2.2 The Beta Function
2.3 The Error Function
2.4 The Incomplete Gamma and Beta Functions
2.5 Practice Problems
3 Orthogonal Polynomials
3.1 Orthogonality
3.2 Legendre Polynomials
3.3 Jacobi Polynomials
3.4 Chebyshev Polynomials of the First Kind
3.5 Chebyshev Polynomials of the Second Kind
3.6 Hermite Polynomials
3.7 Laguerre Polynomials
3.8 Applications of Orthogonal Polynomials
3.9 Practice Problems
4 Sturm–Liouville Theory
4.1 The Sturm–Liouville Problem
4.2 Reduction to Sturm–Liouville Form
4.3 Orthogonality: the Central Theorem
4.4 The Classical Families as Sturm–Liouville Problems
4.5 Eigenfunction Expansions
4.6 Bessel’s Equation
4.7 Practice Problems
5 Fourier Series Analysis
5.1 The Series and Its Coefficients
5.2 Periodicity and Periodic Extension
5.3 Worked Examples
5.4 The Dirichlet Kernel
5.5 Convergence of Fourier Series
5.6 Symmetry and Half-Range Expansions
5.7 Parseval’s Identity
5.8 The Complex Form
5.9 From Fourier Series to the Fourier Transform
5.10 Practice Problems
6 The Laplace Transform
6.1 Definition and Existence
6.2 Operational Properties
6.3 Solving Initial-Value Problems
6.4 Inversion
6.5 Practice Problems
7 Asymptotic Expansions
7.1 Asymptotic Sequences and Expansions
7.2 Watson’s Lemma
7.3 Laplace’s Method
7.4 Stirling’s Formula
7.5 Practice Problems