Sources

The editions this handbook is written from. Page numbers in citations are the printed page numbers of these editions.

1. Lars V. Ahlfors (1979). Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable. 3rd ed. New York: McGraw-Hill.

   Cited by: Convergence tests for series, Absolute and conditional convergence, Uniform convergence, Power series, Topological and metric spaces, Connectedness and simple connectivity, Complex numbers and elementary functions, Analyticity and the Cauchy-Riemann equations, Branch points and branch cuts, Contour integrals, Cauchy's theorem, Cauchy's integral formula, Liouville's theorem and the maximum modulus principle, Taylor series of analytic functions, Laurent series, Classification of isolated singularities, Analytic continuation, The residue theorem, Trigonometric integrals by residues, Infinite real integrals and Jordan's lemma, Integrals around branch cuts, Principal values and indented contours, Summing series with residues, Argument principle and Rouche's theorem, The winding number, Conformal mappings, Pole expansions and infinite products, Riemann surfaces, Riemann zeta function and Bernoulli numbers, The hypergeometric function, Elliptic integrals, Asymptotic series, Euler-Maclaurin summation, Stirling's formula, Partial fractions for inverse transforms, Harmonic functions, Poisson's integral formula

2. George B. Arfken, Hans J. Weber, and Frank E. Harris (2013). Mathematical Methods for Physicists: A Comprehensive Guide. 7th ed. Waltham, MA: Academic Press (Elsevier). ISBN 978-0-12-384654-9.

   Cited by: Convergence tests for series, Absolute and conditional convergence, Uniform convergence, Integrating and differentiating series term by term, Power series, Taylor's theorem, Implicit and inverse function theorems, Lagrange multipliers, Multiple integrals and change of variables, Differentiation under the integral sign, Gaussian integrals, Connectedness and simple connectivity, Determinants, Linear systems and rank, Orthonormal bases and Gram-Schmidt, Eigenvalues and eigenvectors, Spectral theorem for Hermitian matrices, Unitary and orthogonal matrices, Normal matrices and simultaneous diagonalization, Generalized eigenproblems and normal modes, Functions of matrices and the matrix exponential, Matrix norms and condition numbers, Vector algebra and index notation, Gradient and directional derivative, Divergence, curl and the Laplacian, Vector differential identities, Line integrals and conservative fields, Surface integrals and flux, The divergence theorem, Stokes' theorem and Green's theorem in the plane, Green's identities, Helmholtz decomposition and potentials, The Dirac delta in calculations, Orthogonal curvilinear coordinates, Cylindrical and spherical coordinates, Complex numbers and elementary functions, Analyticity and the Cauchy-Riemann equations, Branch points and branch cuts, Contour integrals, Cauchy's theorem, Cauchy's integral formula, Liouville's theorem and the maximum modulus principle, Taylor series of analytic functions, Laurent series, Classification of isolated singularities, Analytic continuation, The residue theorem, Trigonometric integrals by residues, Infinite real integrals and Jordan's lemma, Integrals around branch cuts, Principal values and indented contours, Summing series with residues, Argument principle and Rouche's theorem, Dispersion relations, Pole expansions and infinite products, Riemann surfaces, Solving first-order equations, Structure of linear equations and the Wronskian, Constant-coefficient equations, Variation of parameters, Reduction of order, The damped and driven harmonic oscillator, Power-series solutions at ordinary points, The Frobenius method, Classifying singular points, including infinity, Solving initial-value problems with the Laplace transform, Adjoint and self-adjoint differential operators, Sturm-Liouville eigenvalue problems, Eigenfunction expansions, The Gamma function, The Beta function, Digamma and polygamma functions, Riemann zeta function and Bernoulli numbers, Error function and incomplete Gamma function, General properties of orthogonal polynomials, Legendre polynomials, Associated Legendre functions, Spherical harmonics, Hermite polynomials, Laguerre polynomials, Chebyshev polynomials, Legendre functions of the second kind, Bessel functions of the first kind, Neumann and Hankel functions, Orthogonality and Fourier-Bessel series, Modified Bessel functions, Spherical Bessel functions, Asymptotic forms of Bessel functions, Integral representations of Bessel functions, The hypergeometric function, Confluent hypergeometric functions, Elliptic integrals, Asymptotic series, Euler-Maclaurin summation, Watson's lemma, The method of steepest descents, Stirling's formula, The WKB approximation, Fourier series, Pointwise convergence of Fourier series, The Gibbs phenomenon, Mean-square convergence and Parseval's identity, Smoothness and decay of Fourier coefficients, The Fourier transform, The convolution theorem, Plancherel's theorem, The uncertainty principle, Fourier transform in several dimensions, The Laplace transform, Partial fractions for inverse transforms, Hankel and Mellin transforms, The discrete Fourier transform and the FFT, Delta-function identities, The Heaviside step and related functions, Classification of second-order equations, First-order equations and characteristics, Types of boundary and initial condition, Travelling-wave solutions, The one-dimensional wave equation, Standing waves and normal modes, Fundamental solution of Laplace's equation, Separation of variables in standard coordinates, Multipole expansions, The Helmholtz equation, Green's function for a boundary-value problem, Eigenfunction expansion of Green's functions, Causal Green's functions for initial-value problems, Green's functions for Laplace and Poisson problems, The method of images, Green's functions for the Helmholtz equation, Green's functions for the wave and heat equations, Scattering and the Born approximation, Classification of integral equations, Neumann series, Separable kernels, Hilbert-Schmidt theory, Volterra equations, Functionals and the first variation, The Euler-Lagrange equation, First integrals and the Beltrami identity, Several functions and higher derivatives, Several independent variables, Isoperimetric problems, Pointwise constraints, Hamilton's principle, The Rayleigh-Ritz method, Adjoints, self-adjoint and unitary operators, Symmetric versus self-adjoint operators, Stone's theorem, The mathematical framework of quantum mechanics, Probability spaces, Random variables and distributions, Expectation, variance and covariance, Joint distributions and transformations, Generating functions, Markov, Chebyshev, Jensen and Chernoff inequalities, Laws of large numbers, Sampling distributions, Least-squares fitting, Propagation of uncertainty, Confidence intervals, Hypothesis testing, The periodogram and spectral estimation, Linear filters, Groups and subgroups, Homomorphisms and quotient groups, The symmetric group, Point groups and crystal symmetry, Quaternions, Representations of groups, Orthogonality relations and characters, Character tables, Product representations and selection rules, Matrix Lie groups and Lie algebras, The rotation group and SU(2), Angular momentum algebra, Addition of angular momenta, Rotation matrices of angular momentum, Spherical tensors and the Wigner-Eckart theorem, The Lorentz group, The classical matrix groups, Casimir operators, Clifford algebras and spinors, Tensors and their transformation laws, The metric tensor, Christoffel symbols and the covariant derivative, Differential operators in general coordinates, Pseudotensors and the Levi-Civita tensor, Stress and strain tensors, Differential forms, The exterior derivative, Integration of forms and the general Stokes theorem, The Hodge star and Maxwell's equations, Closed and exact forms, Geodesics, The Riemann curvature tensor, Floating-point arithmetic and error, Computing eigenvalues, Spectral methods

3. Earl A. Coddington (1989). An Introduction to Ordinary Differential Equations. Dover ed. (corrected republication of the 1961 Prentice-Hall ed.). New York: Dover. ISBN 978-0-486-65942-8.

   Cited by: Solving first-order equations, Existence and uniqueness of solutions, Structure of linear equations and the Wronskian, Constant-coefficient equations, Variation of parameters, Reduction of order, Euler-Cauchy equations, Power-series solutions at ordinary points, The Frobenius method, Classifying singular points, including infinity, Linear systems and the matrix exponential, Nonhomogeneous systems and Duhamel's principle, Floquet theory, Sturm comparison and oscillation theorems, The hypergeometric function

4. Manfredo P. do Carmo (2016). Differential Geometry of Curves and Surfaces. Revised and updated 2nd ed. Mineola, NY: Dover. ISBN 978-0-486-80699-0.

   Cited by: Partial derivatives and the chain rule, Continuity and homeomorphism, Connectedness and simple connectivity, Surface integrals and flux, Sturm comparison and oscillation theorems, First integrals and the Beltrami identity, Several independent variables, Pointwise constraints, Conjugate points and Jacobi's condition, Christoffel symbols and the covariant derivative, Curves and the Frenet-Serret formulas, Regular surfaces and the first fundamental form, The second fundamental form and curvature, Gauss's Theorema Egregium, Geodesics, The Gauss-Bonnet theorem, Riemannian manifolds, Pushforwards, pullbacks and immersions, The exponential map, Killing vector fields and symmetries, Index theory

5. David S. Dummit and Richard M. Foote (2004). Abstract Algebra. 3rd ed. Hoboken, NJ: Wiley. ISBN 978-0-471-43334-7. Note: The text layer is a weak OCR; prose is usable, formulas are not.

   Cited by: Zorn's lemma and the axiom of choice, Vector spaces and bases, Determinants, Jordan normal form, Linear systems and the matrix exponential, Groups and subgroups, Cosets and Lagrange's theorem, Homomorphisms and quotient groups, Group actions, Conjugacy classes, The symmetric group, Point groups and crystal symmetry, Rings and fields, Quaternions, Representations of groups, Schur's lemma, Orthogonality relations and characters, Character tables, Product representations and selection rules, The classical matrix groups, Differential forms

6. Lawrence C. Evans (2010). Partial Differential Equations. Graduate Studies in Mathematics 19. 2nd ed. Providence, RI: American Mathematical Society. ISBN 978-0-8218-4974-3.

   Cited by: Lagrange multipliers, The divergence theorem, Green's identities, Nonhomogeneous systems and Duhamel's principle, Airy functions, Laplace's method, The method of stationary phase, Singular perturbations and boundary layers, The Laplace transform, Classification of second-order equations, First-order equations and characteristics, Conservation laws and shocks, Types of boundary and initial condition, Similarity and self-similar solutions, Travelling-wave solutions, The one-dimensional wave equation, Energy and uniqueness for the wave equation, The wave equation in three dimensions, The heat kernel, Maximum principle for the heat equation, Advection, diffusion and reaction, Harmonic functions, Fundamental solution of Laplace's equation, Poisson's integral formula, Separation of variables in standard coordinates, The Helmholtz equation, Weak derivatives and Sobolev spaces, The Lax-Milgram theorem and the Dirichlet problem, Energy estimates and well-posedness, Evolution equations and operator semigroups, The Radon transform, The Fredholm alternative and modified Green's functions, Green's functions for Laplace and Poisson problems, The method of images, Green's functions for the wave and heat equations, Functionals and the first variation, Several independent variables, Noether's theorem, The canonical equations, The Hamilton-Jacobi equation, The second variation and Legendre's condition, Sigma-algebras and measures, Lebesgue measure, Measurable functions, The Lebesgue integral, Monotone convergence and Fatou's lemma, Dominated convergence, Modes of convergence, The contraction mapping theorem, Normed and Banach spaces, The Riesz representation theorem, Weak convergence, Bounded linear operators, Spectrum and resolvent, Compact operators, Spectral theorem for compact self-adjoint operators, Uniform boundedness and open mapping theorems, Unbounded and closed operators, Markov, Chebyshev, Jensen and Chernoff inequalities, Large deviations and Cramer's theorem, The Fokker-Planck equation, Stress and strain tensors, The second fundamental form and curvature, Partitions of unity, The finite element method

7. Gerald B. Folland (1999). Real Analysis: Modern Techniques and Their Applications. 2nd ed. New York: Wiley. ISBN 978-0-471-31716-6.

   Cited by: Limits of sequences and completeness, Taylor's theorem, Implicit and inverse function theorems, Multiple integrals and change of variables, Sets, functions and countability, Zorn's lemma and the axiom of choice, Topological and metric spaces, Continuity and homeomorphism, Compactness, Determinants, The Gamma function, Stirling's formula, Pointwise convergence of Fourier series, The Gibbs phenomenon, Cesàro means and Fejér's theorem, The Fourier transform, Fourier inversion, The convolution theorem, Plancherel's theorem, Poisson summation formula, The sampling theorem, Test functions and distributions, Fourier transforms of distributions, Delta-function identities, Sigma-algebras and measures, Lebesgue measure, Measurable functions, The Lebesgue integral, Monotone convergence and Fatou's lemma, Dominated convergence, Fubini and Tonelli theorems, Modes of convergence, L^p spaces, The Radon-Nikodym theorem, Metric spaces and completeness, Normed and Banach spaces, Hilbert spaces, The projection theorem, Orthonormal bases in Hilbert space, The Riesz representation theorem, Weak convergence, Uniform boundedness and open mapping theorems, The Hahn-Banach theorem, The spectral theorem, Probability spaces, Random variables and distributions, Expectation, variance and covariance, Joint distributions and transformations, Conditional expectation, Characteristic functions, Convergence of random variables, Laws of large numbers, The central limit theorem, Brownian motion, Partitions of unity, Fractals and dimension

8. I. M. Gelfand and S. V. Fomin (1963). Calculus of Variations. Revised English ed., trans. and ed. R. A. Silverman. Englewood Cliffs, NJ: Prentice-Hall. Note: Pre-ISBN printing. The Dover reprint (2000, ISBN 978-0-486-41448-5) has the same text.

   Cited by: Lagrange multipliers, Quadratic forms and the Rayleigh quotient, The Klein-Gordon equation, Functionals and the first variation, The Euler-Lagrange equation, First integrals and the Beltrami identity, Several functions and higher derivatives, Several independent variables, Free endpoints and natural boundary conditions, Isoperimetric problems, Pointwise constraints, Hamilton's principle, Noether's theorem, The canonical equations, The Hamilton-Jacobi equation, The second variation and Legendre's condition, Conjugate points and Jacobi's condition, Variational characterization of eigenvalues, The Rayleigh-Ritz method, The second fundamental form and curvature, Geodesics, Symplectic manifolds and Hamiltonian vector fields

9. Geoffrey R. Grimmett and David R. Stirzaker (2020). Probability and Random Processes. 4th ed. Oxford: Oxford University Press. ISBN 978-0-19-884759-5.

   Cited by: Jordan normal form, The Beta function, Riemann zeta function and Bernoulli numbers, Watson's lemma, Laplace's method, Stirling's formula, Advection, diffusion and reaction, Evolution equations and operator semigroups, Sigma-algebras and measures, The Lebesgue integral, Monotone convergence and Fatou's lemma, Dominated convergence, Modes of convergence, L^p spaces, The Radon-Nikodym theorem, The projection theorem, Probability spaces, Random variables and distributions, Expectation, variance and covariance, Joint distributions and transformations, Conditional expectation, The multivariate normal distribution, Generating functions, Characteristic functions, Markov, Chebyshev, Jensen and Chernoff inequalities, Convergence of random variables, Laws of large numbers, The central limit theorem, Large deviations and Cramer's theorem, Discrete-time Markov chains, Convergence of Markov chains, Random walks, The Poisson process, Continuous-time Markov chains, Stationary processes and spectral densities, Brownian motion, Martingales, The Ito integral and Ito's formula, Stochastic differential equations, The Fokker-Planck equation, Sampling distributions, Maximum likelihood, Bayesian inference, Principal components and empirical orthogonal functions, The periodogram and spectral estimation, Linear filters, Autoregressive models and red noise, The Kalman filter, Optimal interpolation and variational data assimilation, Monte Carlo methods

10. Sadri Hassani (2013). Mathematical Physics: A Modern Introduction to Its Foundations. 2nd ed. Cham: Springer. ISBN 978-3-319-01194-3.

    Cited by: Absolute and conditional convergence, Power series, Gaussian integrals, Sets, functions and countability, Continuity and homeomorphism, Vector spaces and bases, Linear maps and matrices, Inner products and norms, Orthonormal bases and Gram-Schmidt, Orthogonal projection and least squares, Eigenvalues and eigenvectors, Spectral theorem for Hermitian matrices, Unitary and orthogonal matrices, Normal matrices and simultaneous diagonalization, Jordan normal form, Quadratic forms and the Rayleigh quotient, Divergence, curl and the Laplacian, The Dirac delta in calculations, Branch points and branch cuts, Cauchy's integral formula, Liouville's theorem and the maximum modulus principle, Taylor series of analytic functions, Laurent series, Classification of isolated singularities, Analytic continuation, The residue theorem, Trigonometric integrals by residues, Infinite real integrals and Jordan's lemma, Integrals around branch cuts, Principal values and indented contours, Conformal mappings, Dispersion relations, Pole expansions and infinite products, Riemann surfaces, Solving first-order equations, Existence and uniqueness of solutions, Dependence on initial data and Gronwall's inequality, Structure of linear equations and the Wronskian, Constant-coefficient equations, Reduction of order, The damped and driven harmonic oscillator, Power-series solutions at ordinary points, The Frobenius method, Classifying singular points, including infinity, Adjoint and self-adjoint differential operators, Sturm-Liouville eigenvalue problems, Eigenfunction expansions, Sturm comparison and oscillation theorems, The Gamma function, The Beta function, Digamma and polygamma functions, Error function and incomplete Gamma function, General properties of orthogonal polynomials, Legendre polynomials, Associated Legendre functions, Spherical harmonics, Hermite polynomials, Laguerre polynomials, Chebyshev polynomials, Legendre functions of the second kind, Bessel functions of the first kind, Neumann and Hankel functions, Orthogonality and Fourier-Bessel series, Modified Bessel functions, Spherical Bessel functions, Asymptotic forms of Bessel functions, Integral representations of Bessel functions, The hypergeometric function, Confluent hypergeometric functions, Airy functions, Laplace's method, The method of steepest descents, Stirling's formula, The WKB approximation, The Gibbs phenomenon, The Fourier transform, Fourier inversion, Fourier transform in several dimensions, Hankel and Mellin transforms, The discrete Fourier transform and the FFT, Test functions and distributions, Delta-function identities, Classification of second-order equations, First-order equations and characteristics, Types of boundary and initial condition, Standing waves and normal modes, Separation of variables in standard coordinates, The Helmholtz equation, The Schrodinger equation, The Klein-Gordon equation, Energy estimates and well-posedness, Green's function for a boundary-value problem, The Fredholm alternative and modified Green's functions, Eigenfunction expansion of Green's functions, Green's functions for Laplace and Poisson problems, The method of images, Green's functions for the Helmholtz equation, Green's functions for the wave and heat equations, Classification of integral equations, Neumann series, Separable kernels, Hilbert-Schmidt theory, Volterra equations, Metric spaces and completeness, Hilbert spaces, The projection theorem, Orthonormal bases in Hilbert space, The Riesz representation theorem, Bounded linear operators, Adjoints, self-adjoint and unitary operators, Spectrum and resolvent, Compact operators, Spectral theorem for compact self-adjoint operators, Unbounded and closed operators, Symmetric versus self-adjoint operators, Groups and subgroups, Cosets and Lagrange's theorem, Homomorphisms and quotient groups, Group actions, Conjugacy classes, The symmetric group, Quaternions, Representations of groups, Schur's lemma, Orthogonality relations and characters, Character tables, Product representations and selection rules, Matrix Lie groups and Lie algebras, The rotation group and SU(2), Angular momentum algebra, Addition of angular momenta, Rotation matrices of angular momentum, Spherical tensors and the Wigner-Eckart theorem, The Lorentz group, The classical matrix groups, Casimir operators, Clifford algebras and spinors, Tensors and their transformation laws, The metric tensor, Differential forms, The exterior derivative, Integration of forms and the general Stokes theorem, The Hodge star and Maxwell's equations, Closed and exact forms, Smooth manifolds, Tangent vectors and vector fields, Pushforwards, pullbacks and immersions, Flows and integral curves, The exponential map, Symplectic manifolds and Hamiltonian vector fields, Computing eigenvalues, Symplectic integrators

11. Morris W. Hirsch, Stephen Smale, and Robert L. Devaney (2013). Differential Equations, Dynamical Systems, and an Introduction to Chaos. 3rd ed. Waltham, MA: Academic Press (Elsevier). ISBN 978-0-12-382010-5.

    Cited by: Jordan normal form, Functions of matrices and the matrix exponential, Gradient and directional derivative, Existence and uniqueness of solutions, Dependence on initial data and Gronwall's inequality, Variation of parameters, The damped and driven harmonic oscillator, Linear systems and the matrix exponential, Nonhomogeneous systems and Duhamel's principle, Flows and integral curves, Flows on the line, Bifurcations in one dimension, Classification of linear planar systems, Linearization and the Hartman-Grobman theorem, Lyapunov functions, Conservative and reversible systems, Limit cycles and the Poincare-Bendixson theorem, The Hopf bifurcation, Relaxation oscillations and averaging, Global bifurcations of cycles, The Lorenz equations, Lyapunov exponents and sensitive dependence, One-dimensional maps, Period doubling and universality, Poincare maps, Integrators for ODEs

12. Erwin Kreyszig (1978). Introductory Functional Analysis with Applications. 1st ed. New York: Wiley.

    Cited by: Limits of sequences and completeness, Zorn's lemma and the axiom of choice, Topological and metric spaces, Continuity and homeomorphism, Compactness, Vector spaces and bases, Linear maps and matrices, Inner products and norms, Orthonormal bases and Gram-Schmidt, Orthogonal projection and least squares, General properties of orthogonal polynomials, Legendre polynomials, Hermite polynomials, Laguerre polynomials, Chebyshev polynomials, Cesàro means and Fejér's theorem, The uncertainty principle, The Schrodinger equation, The Fredholm alternative and modified Green's functions, Classification of integral equations, Neumann series, Separable kernels, Volterra equations, The Lebesgue integral, Metric spaces and completeness, The contraction mapping theorem, Normed and Banach spaces, Hilbert spaces, The projection theorem, Orthonormal bases in Hilbert space, The Riesz representation theorem, Weak convergence, Bounded linear operators, Adjoints, self-adjoint and unitary operators, Spectrum and resolvent, Compact operators, Spectral theorem for compact self-adjoint operators, Uniform boundedness and open mapping theorems, The Hahn-Banach theorem, Unbounded and closed operators, Symmetric versus self-adjoint operators, The spectral theorem, The mathematical framework of quantum mechanics, Root finding, Polynomial and spline interpolation, Numerical quadrature, Solving linear systems numerically, Computing eigenvalues, Spectral methods

13. John M. Lee (2013). Introduction to Smooth Manifolds. Graduate Texts in Mathematics 218. 2nd ed. New York: Springer. ISBN 978-1-4419-9981-8.

    Cited by: Partial derivatives and the chain rule, Taylor expansion and critical points in several variables, Implicit and inverse function theorems, Lagrange multipliers, Multiple integrals and change of variables, Differentiation under the integral sign, Sets, functions and countability, Zorn's lemma and the axiom of choice, Topological and metric spaces, Continuity and homeomorphism, Compactness, Connectedness and simple connectivity, Vector spaces and bases, Linear maps and matrices, Determinants, Linear systems and rank, Inner products and norms, Orthonormal bases and Gram-Schmidt, Orthogonal projection and least squares, Jordan normal form, Gradient and directional derivative, Line integrals and conservative fields, The divergence theorem, Stokes' theorem and Green's theorem in the plane, Orthogonal curvilinear coordinates, The canonical equations, Fubini and Tonelli theorems, Stone's theorem, Group actions, Quaternions, Matrix Lie groups and Lie algebras, The classical matrix groups, The metric tensor, Differential operators in general coordinates, Differential forms, The exterior derivative, Integration of forms and the general Stokes theorem, The Hodge star and Maxwell's equations, Closed and exact forms, Regular surfaces and the first fundamental form, The Gauss-Bonnet theorem, Smooth manifolds, Tangent vectors and vector fields, Riemannian manifolds, Tangent and vector bundles, Pushforwards, pullbacks and immersions, Flows and integral curves, The exponential map, Partitions of unity, Symplectic manifolds and Hamiltonian vector fields, Killing vector fields and symmetries, Symplectic integrators

14. Michael Reed and Barry Simon (1980). Methods of Modern Mathematical Physics, Vol. I: Functional Analysis. Revised and enlarged ed. New York: Academic Press. ISBN 978-0-12-585050-6. Note: The scan has no title or copyright page. The edition is inferred from the preface dated April 1980; the ISBN is that of the revised and enlarged edition and is likewise inferred.

    Cited by: Limits of sequences and completeness, Zorn's lemma and the axiom of choice, Topological and metric spaces, Continuity and homeomorphism, Inner products and norms, Orthonormal bases and Gram-Schmidt, Singular value decomposition, Dispersion relations, Adjoint and self-adjoint differential operators, Hermite polynomials, Mean-square convergence and Parseval's identity, The Fourier transform, Fourier inversion, The convolution theorem, Plancherel's theorem, Fourier transforms of distributions, The Heaviside step and related functions, The Schrodinger equation, The Klein-Gordon equation, Classification of integral equations, Hilbert-Schmidt theory, Sigma-algebras and measures, Lebesgue measure, Measurable functions, The Lebesgue integral, Monotone convergence and Fatou's lemma, Dominated convergence, Fubini and Tonelli theorems, L^p spaces, The Radon-Nikodym theorem, Metric spaces and completeness, The contraction mapping theorem, Normed and Banach spaces, Hilbert spaces, The projection theorem, The Riesz representation theorem, Weak convergence, Bounded linear operators, Adjoints, self-adjoint and unitary operators, Spectrum and resolvent, Compact operators, Spectral theorem for compact self-adjoint operators, Uniform boundedness and open mapping theorems, The Hahn-Banach theorem, Unbounded and closed operators, Symmetric versus self-adjoint operators, The spectral theorem, Stone's theorem, The mathematical framework of quantum mechanics, Characteristic functions

15. K. F. Riley, M. P. Hobson, and S. J. Bence (2006). Mathematical Methods for Physics and Engineering: A Comprehensive Guide. 3rd ed. Cambridge: Cambridge University Press. ISBN 978-0-521-67971-8.

    Cited by: Convergence tests for series, Absolute and conditional convergence, Uniform convergence, Integrating and differentiating series term by term, Power series, Taylor's theorem, Partial derivatives and the chain rule, Taylor expansion and critical points in several variables, Lagrange multipliers, Multiple integrals and change of variables, Gaussian integrals, Connectedness and simple connectivity, Vector spaces and bases, Linear maps and matrices, Determinants, Linear systems and rank, Inner products and norms, Orthonormal bases and Gram-Schmidt, Eigenvalues and eigenvectors, Spectral theorem for Hermitian matrices, Unitary and orthogonal matrices, Normal matrices and simultaneous diagonalization, Quadratic forms and the Rayleigh quotient, Generalized eigenproblems and normal modes, Singular value decomposition, Functions of matrices and the matrix exponential, Vector algebra and index notation, Gradient and directional derivative, Divergence, curl and the Laplacian, Vector differential identities, Line integrals and conservative fields, Surface integrals and flux, The divergence theorem, Stokes' theorem and Green's theorem in the plane, Green's identities, Helmholtz decomposition and potentials, The Dirac delta in calculations, Orthogonal curvilinear coordinates, Cylindrical and spherical coordinates, Complex numbers and elementary functions, Analyticity and the Cauchy-Riemann equations, Branch points and branch cuts, Contour integrals, Cauchy's theorem, Cauchy's integral formula, Liouville's theorem and the maximum modulus principle, Taylor series of analytic functions, Laurent series, Classification of isolated singularities, The residue theorem, Trigonometric integrals by residues, Infinite real integrals and Jordan's lemma, Integrals around branch cuts, Principal values and indented contours, Summing series with residues, Argument principle and Rouche's theorem, Conformal mappings, Solving first-order equations, Structure of linear equations and the Wronskian, Constant-coefficient equations, Variation of parameters, Reduction of order, Euler-Cauchy equations, Power-series solutions at ordinary points, The Frobenius method, Classifying singular points, including infinity, Solving initial-value problems with the Laplace transform, Adjoint and self-adjoint differential operators, Sturm-Liouville eigenvalue problems, Eigenfunction expansions, The Gamma function, The Beta function, Riemann zeta function and Bernoulli numbers, Error function and incomplete Gamma function, General properties of orthogonal polynomials, Legendre polynomials, Associated Legendre functions, Spherical harmonics, Hermite polynomials, Laguerre polynomials, Chebyshev polynomials, Legendre functions of the second kind, Bessel functions of the first kind, Neumann and Hankel functions, Orthogonality and Fourier-Bessel series, Spherical Bessel functions, Asymptotic forms of Bessel functions, Integral representations of Bessel functions, The hypergeometric function, Confluent hypergeometric functions, Airy functions, Asymptotic series, Watson's lemma, Laplace's method, The method of stationary phase, The method of steepest descents, Regular perturbation expansions, The WKB approximation, Fourier series, Pointwise convergence of Fourier series, Mean-square convergence and Parseval's identity, The Fourier transform, Fourier inversion, The convolution theorem, Plancherel's theorem, The uncertainty principle, Poisson summation formula, Fourier transform in several dimensions, The Laplace transform, Partial fractions for inverse transforms, Hankel and Mellin transforms, The discrete Fourier transform and the FFT, The sampling theorem, Delta-function identities, The Heaviside step and related functions, Classification of second-order equations, Types of boundary and initial condition, The one-dimensional wave equation, Standing waves and normal modes, Fundamental solution of Laplace's equation, Separation of variables in standard coordinates, Multipole expansions, The Helmholtz equation, The Schrodinger equation, Green's function for a boundary-value problem, Eigenfunction expansion of Green's functions, Causal Green's functions for initial-value problems, Green's functions for Laplace and Poisson problems, The method of images, Green's functions for the Helmholtz equation, Scattering and the Born approximation, Classification of integral equations, Neumann series, Separable kernels, Hilbert-Schmidt theory, Volterra equations, The Euler-Lagrange equation, First integrals and the Beltrami identity, Several functions and higher derivatives, Free endpoints and natural boundary conditions, Isoperimetric problems, Variational characterization of eigenvalues, The Rayleigh-Ritz method, Orthonormal bases in Hilbert space, Probability spaces, Random variables and distributions, Expectation, variance and covariance, Joint distributions and transformations, Conditional expectation, The multivariate normal distribution, Generating functions, Characteristic functions, The central limit theorem, Random walks, The Poisson process, Continuous-time Markov chains, Stationary processes and spectral densities, Sampling distributions, Estimators and their properties, Maximum likelihood, Least-squares fitting, Propagation of uncertainty, Confidence intervals, Hypothesis testing, Bayesian inference, Principal components and empirical orthogonal functions, The periodogram and spectral estimation, Linear filters, The Kalman filter, Optimal interpolation and variational data assimilation, Groups and subgroups, Cosets and Lagrange's theorem, Homomorphisms and quotient groups, Conjugacy classes, The symmetric group, Point groups and crystal symmetry, Representations of groups, Orthogonality relations and characters, Character tables, Product representations and selection rules, The rotation group and SU(2), Angular momentum algebra, Rotation matrices of angular momentum, Clifford algebras and spinors, Tensors and their transformation laws, The metric tensor, Christoffel symbols and the covariant derivative, Differential operators in general coordinates, Pseudotensors and the Levi-Civita tensor, Stress and strain tensors, Curves and the Frenet-Serret formulas, Geodesics, The Riemann curvature tensor, Floating-point arithmetic and error, Root finding, Numerical quadrature, Solving linear systems numerically, Computing eigenvalues, Integrators for ODEs, Stability and stiff equations, Finite-difference approximations, Spectral methods, Monte Carlo methods

16. Ivar Stakgold and Michael Holst (2011). Green's Functions and Boundary Value Problems. 3rd ed. Hoboken, NJ: Wiley. ISBN 978-0-470-60970-5.

    Cited by: Integrating and differentiating series term by term, Taylor's theorem, Differentiation under the integral sign, Vector spaces and bases, Linear systems and rank, Inner products and norms, Orthonormal bases and Gram-Schmidt, Orthogonal projection and least squares, Singular value decomposition, Matrix norms and condition numbers, The Dirac delta in calculations, Structure of linear equations and the Wronskian, Adjoint and self-adjoint differential operators, Sturm-Liouville eigenvalue problems, Eigenfunction expansions, Error function and incomplete Gamma function, Chebyshev polynomials, Neumann and Hankel functions, Stirling's formula, Regular perturbation expansions, Singular perturbations and boundary layers, Fourier series, Pointwise convergence of Fourier series, The Gibbs phenomenon, Cesàro means and Fejér's theorem, Mean-square convergence and Parseval's identity, Smoothness and decay of Fourier coefficients, The Fourier transform, Fourier inversion, The convolution theorem, Plancherel's theorem, Poisson summation formula, Hankel and Mellin transforms, The discrete Fourier transform and the FFT, The sampling theorem, Test functions and distributions, Fourier transforms of distributions, Delta-function identities, The Heaviside step and related functions, Classification of second-order equations, First-order equations and characteristics, Types of boundary and initial condition, Similarity and self-similar solutions, The heat kernel, Maximum principle for the heat equation, Diffusion in bounded domains, Advection, diffusion and reaction, Harmonic functions, Fundamental solution of Laplace's equation, Poisson's integral formula, Multipole expansions, Weak derivatives and Sobolev spaces, The Lax-Milgram theorem and the Dirichlet problem, Energy estimates and well-posedness, Green's function for a boundary-value problem, The Fredholm alternative and modified Green's functions, Eigenfunction expansion of Green's functions, Causal Green's functions for initial-value problems, Classification of integral equations, Neumann series, Separable kernels, Hilbert-Schmidt theory, Volterra equations, Free endpoints and natural boundary conditions, The Rayleigh-Ritz method, Measurable functions, The Lebesgue integral, Monotone convergence and Fatou's lemma, Dominated convergence, Fubini and Tonelli theorems, L^p spaces, Metric spaces and completeness, The contraction mapping theorem, Normed and Banach spaces, Hilbert spaces, The projection theorem, Orthonormal bases in Hilbert space, The Riesz representation theorem, Weak convergence, Bounded linear operators, Adjoints, self-adjoint and unitary operators, Spectrum and resolvent, Compact operators, Spectral theorem for compact self-adjoint operators, Uniform boundedness and open mapping theorems, The Hahn-Banach theorem, Unbounded and closed operators, Symmetric versus self-adjoint operators, The periodogram and spectral estimation, Christoffel symbols and the covariant derivative, Stress and strain tensors, The Riemann curvature tensor, Riemannian manifolds, Linearization and the Hartman-Grobman theorem, Solving linear systems numerically, Computing eigenvalues, Relaxation methods for elliptic problems, Spectral methods, The finite element method

17. Elias M. Stein and Rami Shakarchi (2003). Fourier Analysis: An Introduction. Princeton Lectures in Analysis, Vol. I. Princeton, NJ: Princeton University Press. ISBN 978-0-691-11384-5.

    Cited by: Pole expansions and infinite products, Riemann zeta function and Bernoulli numbers, Euler-Maclaurin summation, Fourier series, Pointwise convergence of Fourier series, Cesàro means and Fejér's theorem, Mean-square convergence and Parseval's identity, Smoothness and decay of Fourier coefficients, The Fourier transform, Fourier inversion, The convolution theorem, Plancherel's theorem, The uncertainty principle, Poisson summation formula, Fourier transform in several dimensions, Hankel and Mellin transforms, The discrete Fourier transform and the FFT, Travelling-wave solutions, The one-dimensional wave equation, Energy and uniqueness for the wave equation, The wave equation in three dimensions, Standing waves and normal modes, The heat kernel, Poisson's integral formula, The Radon transform

18. Walter A. Strauss (2008). Partial Differential Equations: An Introduction. 2nd ed. Hoboken, NJ: Wiley. ISBN 978-0-470-05456-7.

    Cited by: Matrix norms and condition numbers, Surface integrals and flux, The divergence theorem, Stokes' theorem and Green's theorem in the plane, Green's identities, Helmholtz decomposition and potentials, Cylindrical and spherical coordinates, Euler-Cauchy equations, Nonhomogeneous systems and Duhamel's principle, Solving initial-value problems with the Laplace transform, Adjoint and self-adjoint differential operators, Sturm-Liouville eigenvalue problems, Eigenfunction expansions, Sturm comparison and oscillation theorems, The Gamma function, Error function and incomplete Gamma function, Legendre polynomials, Associated Legendre functions, Spherical harmonics, Bessel functions of the first kind, Neumann and Hankel functions, Orthogonality and Fourier-Bessel series, Spherical Bessel functions, Asymptotic forms of Bessel functions, Airy functions, Fourier series, Pointwise convergence of Fourier series, The Gibbs phenomenon, Smoothness and decay of Fourier coefficients, The Fourier transform, Plancherel's theorem, The uncertainty principle, Poisson summation formula, Fourier transform in several dimensions, The Laplace transform, The sampling theorem, Test functions and distributions, The Heaviside step and related functions, Classification of second-order equations, Conservation laws and shocks, Types of boundary and initial condition, Similarity and self-similar solutions, Travelling-wave solutions, The one-dimensional wave equation, Energy and uniqueness for the wave equation, The wave equation in three dimensions, Standing waves and normal modes, The heat kernel, Maximum principle for the heat equation, Diffusion in bounded domains, Advection, diffusion and reaction, Harmonic functions, Fundamental solution of Laplace's equation, Poisson's integral formula, The Klein-Gordon equation, Energy estimates and well-posedness, Green's functions for Laplace and Poisson problems, The method of images, Green's functions for the Helmholtz equation, Green's functions for the wave and heat equations, Scattering and the Born approximation, Several independent variables, Variational characterization of eigenvalues, The Rayleigh-Ritz method, Brownian motion, Bifurcations in one dimension, Floating-point arithmetic and error, Finite-difference approximations, Stability, the CFL condition and von Neumann analysis, Relaxation methods for elliptic problems, The finite element method

19. Steven H. Strogatz (1994). Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. 1st ed. Reading, MA: Perseus Books. ISBN 0-201-54344-3.

    Cited by: Existence and uniqueness of solutions, Dependence on initial data and Gronwall's inequality, The damped and driven harmonic oscillator, Floquet theory, Elliptic integrals, Regular perturbation expansions, Singular perturbations and boundary layers, Flows on the line, Bifurcations in one dimension, Flows on the circle, Classification of linear planar systems, Linearization and the Hartman-Grobman theorem, Lyapunov functions, Conservative and reversible systems, Index theory, Limit cycles and the Poincare-Bendixson theorem, The Hopf bifurcation, Relaxation oscillations and averaging, Global bifurcations of cycles, The Lorenz equations, Lyapunov exponents and sensitive dependence, One-dimensional maps, Period doubling and universality, Poincare maps, Fractals and dimension, The Henon map and strange attractors, Integrators for ODEs