Hochstadt H. The Functions of Mathematical Physics

Department of Mathematics Polytechnic Institute of New York, 1971, 1986. 150 p. — (ISBN13: 978-0-486-65214-6, ISBN10: 0-486-65214-9).Orthogonal Polynomials.
Linear Spaces.
Orthogonal Polynomials.
The Recurrence Formula.
The Christoffel—Darboux Formula.
The Weierstrass Approximation Theorem.
The Zeros of the Orthogonal Polynomials.
Approximation Theory.
More about the Zeros of the Orthonormal Polynomials.
The completeness of the Orthonormal Polynomials in the Space of Square-Integrable Functions.
Generalizations and an Application to Conformal Mappings.
The Classical Orthogonal Polynomials.
Rodrigues' Formula and the Classical Orthogonal Polynomials.
The Differential Equations Satisfied by the Classical Orthogonal Polynomials.
On the Zeros of the Jacobi Polynomials.
An Alternative Approach to the Tchebicheff Polynomials.
An Application of the Hermite Polynomials to Quantum Mechanics.
The Completeness of the Hermite and Laguerre Polynomials.
Generating Functions.
The Gamma Function.
Definitions and Basic Properties.
Analytic Continuation and Integral Representations.
Asymptotic Expansions.
Beta Functions.
The Logarithmic Derivative of the Gamma Function.
Mellin-Barnes Integrals.
Mellin Transforms.
Applications to Algebraic Equations.
Hypergeometric Functions.
Review of Linear Differential Equations with Regular Singular Points.
The Hypergeometric Differential Equation.
The Hypergeometric Function.
A General Method for Finding Integral Representations.
ntegral Representations for the Hypergeometric Function.
The Twenty-four Solutions of the Hypergeometric Equation.
The Schwarz-Christoffel Transformation.
Mappings of Curvilinear Triangles.
Group Theoretic Discussion of the Case π(α1 + α2 + α3) π.
Nonlinear Transformations of Hypergeometric Functions.
The Legendre Functions.
Laplace's Differential Equation.
Maxwell's Theory of Poles.
Relationship to the Hypergeometric Functions.
Expansion Formulas.
The Addition Theorem.
Green's Functions.
The Complete Solution of Legendre's Differential Equation.
Asymptotic Formulas.
Spherical Harmonics in p Dimensions.
Homogeneous Polynomials.
Orthogonality of Spherical Harmonics.
Legendre Polynomials.
Applications to Boundary Value Problems.
Confluent Hypergeometric Functions.
Relationship to the Hypergeometric Functions.
Applications of These Functions in Mathematical Physics.
Integral Representations.
Asymptotic Representations.
Bessel Functions.
Basic Definitions.
Integral Representations.
Relationship to the Legendre Functions.
The Generating Function of the Bessel Function.
More Integral Representations.
Addition Theorems.
The Complete Solution of Bessel's Equation.
Asymptotic Expansions for Large Argument.
Airy Functions.
Asymptotic Expansions for Large Indices and Large Arguments.
Some Applications of Bessel Functions in Physical Optics.
The Zeros of Bessel Functions.
Fourier-Bessel Expansions.
Applications in Mathematical Physics.
Discontinuous Integrals.
Hill's Equation.
Mathieu's Equation.
Hill's Equation.
The Discriminant.
Expansion Theorems.
Inverse Problems.
Hill's Equations with Even Coefficients.
Mathieu's Equation Revisited.
Energy Bands in Crystals.
Appendix.