Mathematical methods in physics, engineering, and chemistry

Cover
Title Page
Copyright Page
Contents
Preface
Chapter 1 Vectors and linear operators
	1.1 The linearity of physical phenomena
	1.2 Vector spaces
		1.2.1 A word on notation
		1.2.2 Linear independence, bases, and dimensionality
		1.2.3 Subspaces
		1.2.4 Isomorphism of N-dimensional spaces
		1.2.5 Dual spaces
	1.3 Inner products and orthogonality
		1.3.1 Inner products
		1.3.2 The Schwarz inequality
		1.3.3 Vector norms
		1.3.4 Orthonormal bases and the Gram–Schmidt process
		1.3.5 Complete sets of orthonormal vectors
	1.4 Operators and matrices
		1.4.1 Linear operators
		1.4.2 Representing operators with matrices
		1.4.3 Matrix algebra
		1.4.4 Rank and nullity
		1.4.5 Bounded operators
		1.4.6 Inverses
		1.4.7 Change of basis and the similarity transformation
		1.4.8 Adjoints and Hermitian operators
		1.4.9 Determinants and the matrix inverse
		1.4.10 Unitary operators
		1.4.11 The trace of a matrix
	1.5 Eigenvectors and their role in representing operators
		1.5.1 Eigenvectors and eigenvalues
		1.5.2 The eigenproblem for Hermitian and unitary operators
		1.5.3 Diagonalizing matrices
	1.6 Hilbert space: Infinite-dimensional vector space
	Exercises
Chapter 2 Sturm–Liouville theory
	2.1 Second-order differential equations
		2.1.1 Uniqueness and linear independence
		2.1.2 The adjoint operator
		2.1.3 Self-adjoint operator
	2.2 Sturm–Liouville systems
	2.3 The Sturm–Liouville eigenproblem
	2.4 The Dirac delta function
	2.5 Completeness
	2.6 Recap
	Summary
	Exercises
Chapter 3 Partial differential equations
	3.1 A survey of partial differential equations
		3.1.1 The continuity equation
		3.1.2 The diffusion equation
		3.1.3 The free-particle Schr¨odinger equation
		3.1.4 The heat equation
		3.1.5 The inhomogeneous diffusion equation
		3.1.6 Schr¨odinger equation for a particle in a potential field
		3.1.7 The Poisson equation
		3.1.8 The Laplace equation
		3.1.9 The wave equation
		3.1.10 Inhomogeneous wave equation
		3.1.11 Summary of PDEs
	3.2 Separation of variables and the Helmholtz equation
		3.2.1 Rectangular coordinates
		3.2.2 Cylindrical coordinates
		3.2.3 Spherical coordinates
	3.3 The paraxial approximation
	3.4 The three types of linear PDEs
		3.4.1 Hyperbolic PDEs
		3.4.2 Parabolic PDEs
		3.4.3 Elliptic PDEs
	3.5 Outlook
	Summary
	Exercises
Chapter 4 Fourier analysis
	4.1 Fourier series
	4.2 The exponential form of Fourier series
	4.3 General intervals
	4.4 Parseval’s theorem
	4.5 Back to the delta function
	4.6 Fourier transform
	4.7 Convolution integral
	Summary
	Exercises
Chapter 5 Series solutions of ordinary differential equations
	5.1 The Frobenius method
		5.1.1 Power series
		5.1.2 Introductory example
		5.1.3 Ordinary points
		5.1.4 Regular singular points
	5.2 Wronskian method for obtaining a second solution
	5.3 Bessel and Neumann functions
	5.4 Legendre polynomials
	Summary
	Exercises
Chapter 6 Spherical harmonics
	6.1 Properties of the Legendre polynomials, Pl(x)
		6.1.1 Rodrigues formula
		6.1.2 Orthogonality
		6.1.3 Completeness
		6.1.5 Recursion relations
	6.2 Associated Legendre functions, Plm (x)
	6.3 Spherical harmonic functions, Y ml (θ, φ)
	6.4 Addition theorem for Y ml (θ, φ)
	6.5 Laplace equation in spherical coordinates
	Summary
	Exercises
Chapter 7 Bessel functions
	7.1 Small-argument and asymptotic forms
		7.1.1 Limiting forms for small argument
		7.1.3 Hankel functions
	7.2 Properties of the Bessel functions, Jn(x)
		7.2.1 Series associated with the generating function
		7.2.2 Recursion relations
		7.2.3 Integral representation
	7.3 Orthogonality
	7.4 Bessel series
	7.5 The Fourier-Bessel transform
	7.6 Spherical Bessel functions
		7.6.1 Reduction to elementary functions
		7.6.2 Small-argument forms
		7.6.3 Asymptotic forms
		7.6.4 Orthogonality and completeness
	7.7 Expansion of plane waves in spherical harmonics
	Summary
	Exercises
Chapter 8 Complex analysis
	8.1 Complex functions
	8.2 Analytic functions: differentiable in a region
		8.2.1 Continuity, differentiability, and analyticity
		8.2.2 Cauchy–Riemann conditions
		8.2.3 Analytic functions are functions only of z = x + iy
		8.2.4 Useful definitions
	8.3 Contour integrals
	8.4 Integrating analytic functions
	8.5 Cauchy integral formulas
		8.5.1 Derivatives of analytic functions
		8.5.2 Consequences of the Cauchy formulas
	8.6 Taylor and Laurent series
		8.6.1 Taylor series
		8.6.2 The zeros of analytic functions are isolated
		8.6.3 Laurent series
	8.7 Singularities and residues
		8.7.1 Isolated singularities, residue theorem
		8.7.2 Multivalued functions, branch points, and branch cuts
	8.8 Definite integrals
		8.8.1 Integrands containing cos θ and sin θ
		8.8.2 Infinite integrals
		8.8.3 Poles on the contour of integration
	8.9 Meromorphic functions
	8.10 Approximation of integrals
		8.10.1 The method of steepest descent
		8.10.2 The method of stationary phase
	8.11 The analytic signal
		8.11.1 The Hilbert transform
		8.11.2 Paley–Wiener and Titchmarsh theorems
		8.11.3 Is the analytic signal, analytic?
	8.12 The Laplace transform
	Summary
	Exercises
Chapter 9 Inhomogeneous differential equations
	9.1 The method of Green functions
		9.1.1 Boundary conditions
		9.1.2 Reciprocity relation: G(x, x') = G(x', x)
		9.1.3 Matching conditions
		9.1.4 Direct construction of G(x, x')
		9.1.5 Eigenfunction expansions
	9.2 Poisson equation
		9.2.1 Boundary conditions and reciprocity relations
		9.2.2 So, what’s the Green function?
	9.3 Helmholtz equation
		9.3.1 Green function for two-dimensional problems
		9.3.2 Free-space Green function for three dimensions
		9.3.3 Expansion in spherical harmonics
	9.4 Diffusion equation
		9.4.1 Boundary conditions, causality, and reciprocity
		9.4.2 Solution to the diffusion equation
		9.4.3 Free-space Green function
	9.5 Wave equation
	9.6 The Kirchhoff integral theorem
	Summary
	Exercises
Chapter 10 Integral equations
	10.1 Introduction
		10.1.1 Equivalence of integral and differential equations
		10.1.2 Role of coordinate systems in capturing boundary data
	10.2 Classification of integral equations
	10.3 Neumann series
	10.4 Integral transform methods
		10.4.1 Difference kernels
		10.4.2 Fourier kernels
	10.5 Separable kernels
	10.6 Self-adjoint kernels
	10.7 Numerical approaches
		10.7.1 Matrix form
		10.7.2 Measurement space
		10.7.3 The generalized inverse
	Summary
	Exercises
Chapter 11 Tensor analysis
	11.1 Once over lightly: A quick intro to tensors
	11.2 Transformation properties
		11.2.1 The two types of vector: Contravariant and covariant
		11.2.2 Coordinate transformations
		11.2.3 Contravariant vectors and tensors
		11.2.4 Covariant vectors and tensors
		11.2.5 Mixed tensors
		11.2.6 Covariant equations
	11.3 Contraction and the quotient theorem
	11.4 The metric tensor
	11.5 Raising and lowering indices
	11.6 Geometric properties of covariant vectors
	11.7 Relative tensors
	11.8 Tensors as operators
	11.9 Symmetric and antisymmetric tensors
	11.10 The Levi-Civita tensor
	11.11 Pseudotensors
	11.12 Covariant differentiation of tensors
	Summary
	Exercises
A Vector calculus
	A.1 Scalar fields
		A.1.1 The directional derivative
		A.1.2 The gradient
	A.2 Vector fields
		A.2.1 Divergence
		A.2.2 Curl
		A.2.3 The Laplacian
		A.2.4 Vector operator formulae
	A.3 Integration
		A.3.1 Line integrals
		A.3.2 Surface integrals
	A.4 Important integral theorems in vector calculus
		A.4.1 Green’s theorem in the plane
		A.4.2 The divergence theorem
		A.4.3 Stokes’ theorem
		A.4.4 Conservative fields
		A.4.5 The Helmholtz theorem
	A.5 Coordinate systems
		A.5.1 Orthogonal curvilinear coordinates
		A.5.2 Unit vectors
		A.5.4 Differential surface and volume elements
		A.5.5 Transformation of vector components
		A.5.6 Cylindrical coordinates
B Power series
C The gamma function, Γ(x)
	Recursion relation
	Limit formula
	Reflection formula
	Digamma function
D Boundary conditions for Partial Differential Equations
	Summary
References
Index
EULA