BRETT BORDEN, PHD, joined the Faculty of the Naval Postgraduate School in Monterey, CA in 2002, where he is Emeritus Professor of Physics. He served on the editorial board of the journal Inverse Problems from 2002 to 2013. Dr. Borden is a Fellow of The Institute of Physics. JAMES LUSCOMBE, PHD, is a Professor at the Naval Postgraduate School in Monterey, California. He teaches a variety of topics, including general relativity, statistical mechanics, mathematical methods, and quantum computation.

• Preface xi1 Vectors and linear operators 11.1 The linearity of physical phenomena 11.2 Vector spaces 21.3 Inner products and orthogonality 101.4 Operators and matrices 161.5 Eigenvectors and their role in representing operators 361.6 Hilbert space: Infinite-dimensional vector space 43Exercises 472 Sturm–Liouville theory 512.1 Second-order differential equations 522.2 Sturm–Liouville systems 572.3 The Sturm–Liouville eigenproblem 602.4 The Dirac delta function 642.5 Completeness 662.6 Recap 68Summary 68Exercises 693 Partial differential equations 713.1 A survey of partial differential equations 713.2 Separation of variables and the Helmholtz equation 763.3 The paraxial approximation 833.4 The three types of linear PDEs 843.5 Outlook 88Summary 88Exercises 894 Fourier analysis 914.1 Fourier series 914.2 The exponential form of Fourier series 964.3 General intervals 984.4 Parseval’s theorem 1034.5 Back to the delta function 1054.6 Fourier transform 1074.7 Convolution integral 111Summary 115Exercises 1165 Series solutions of ordinary differential equations 1215.1 The Frobenius method 1225.2 Wronskian method for obtaining a second solution 1375.3 Bessel and Neumann functions 1375.4 Legendre polynomials 142Summary 144Exercises 1456 Spherical harmonics 1476.1 Properties of the Legendre polynomials, Pl(x) 1486.2 Associated Legendre functions, Pm l (x) 1576.3 Spherical harmonic functions, Yml (θ, φ) 1586.4 Addition theorem for Ym l (θ, φ) 1606.5 Laplace equation in spherical coordinates 166Summary 167Exercises 1687 Bessel functions 1737.1 Small-argument and asymptotic forms 1737.2 Properties of the Bessel functions, Jn(x) 1757.3 Orthogonality 1807.4 Bessel series 1827.5 The Fourier-Bessel transform 1857.6 Spherical Bessel functions 1867.7 Expansion of plane waves in spherical harmonics 190Summary 192Exercises 1928 Complex analysis 1958.1 Complex functions 1958.2 Analytic functions: differentiable in a region 1978.3 Contour integrals 2028.4 Integrating analytic functions 2068.5 Cauchy integral formulas 2108.6 Taylor and Laurent series 2138.7 Singularities and residues 2178.8 Definite integrals 2218.9 Meromorphic functions 2288.10 Approximation of integrals 2308.11 The analytic signal 2368.12 The Laplace transform 242Summary 245Exercises 2459 Inhomogeneous differential equations 2519.1 The method of Green functions 2519.2 Poisson equation 2609.3 Helmholtz equation 2669.4 Diffusion equation 2729.5 Wave equation 2799.6 The Kirchhoff integral theorem 283Summary 284Exercises 28410 Integral equations 28710.1 Introduction 28710.2 Classification of integral equations 29010.3 Neumann series 29110.4 Integral transform methods 29310.5 Separable kernels 29510.6 Self-adjoint kernels 29710.7 Numerical approaches 302Summary 314Exercises 31511 Tensor analysis 31911.1 Once over lightly: A quick intro to tensors 31911.2 Transformation properties 32711.3 Contraction and the quotient theorem 34011.4 The metric tensor 34211.5 Raising and lowering indices 34411.6 Geometric properties of covariant vectors 34711.7 Relative tensors 35011.8 Tensors as operators 35311.9 Symmetric and antisymmetric tensors 35611.10 The Levi-Civita tensor 35711.11 Pseudotensors 36011.12 Covariant differentiation of tensors 363Summary 373Exercises 374A Vector calculus 377A.1 Scalar fields 377A.1.1 The directional derivative 377A.1.2 The gradient 378A.2 Vector fields 379A.2.1 Divergence 379A.2.2 Curl 380A.2.3 The Laplacian 380A.2.4 Vector operator formulae 381A.3 Integration 382A.3.1 Line integrals 382A.3.2 Surface integrals 383A.4 Important integral theorems in vector calculus 384A.4.1 Green’s theorem in the plane 384A.4.2 The divergence theorem 386A.4.3 Stokes’ theorem 386A.4.4 Conservative fields 387A.4.5 The Helmholtz theorem 389A.5 Coordinate systems 390A.5.1 Orthogonal curvilinear coordinates 390A.5.2 Unit vectors 391A.5.3 Differential displacement 392A.5.4 Differential surface and volume elements 393A.5.5 Transformation of vector components 393A.5.6 Cylindrical coordinates 394B Power series 401C The gamma function, Γ(x) 403Recursion relation 403Limit formula 404Reflection formula 405Digamma function 405D Boundary conditions for Partial Differential Equations 409Summary 417References 419Index 421

BRETT BORDEN, PHD, joined the Faculty of the Naval Postgraduate School in Monterey, CA in 2002, where he is Emeritus Professor of Physics. He served on the editorial board of the journal Inverse Problems from 2002 to 2013. Dr. Borden is a Fellow of The Institute of Physics. JAMES LUSCOMBE, PHD, is a Professor at the Naval Postgraduate School in Monterey, California. He teaches a variety of topics, including general relativity, statistical mechanics, mathematical methods, and quantum computation.

• Preface xi1 Vectors and linear operators 11.1 The linearity of physical phenomena 11.2 Vector spaces 21.3 Inner products and orthogonality 101.4 Operators and matrices 161.5 Eigenvectors and their role in representing operators 361.6 Hilbert space: Infinite-dimensional vector space 43Exercises 472 Sturm–Liouville theory 512.1 Second-order differential equations 522.2 Sturm–Liouville systems 572.3 The Sturm–Liouville eigenproblem 602.4 The Dirac delta function 642.5 Completeness 662.6 Recap 68Summary 68Exercises 693 Partial differential equations 713.1 A survey of partial differential equations 713.2 Separation of variables and the Helmholtz equation 763.3 The paraxial approximation 833.4 The three types of linear PDEs 843.5 Outlook 88Summary 88Exercises 894 Fourier analysis 914.1 Fourier series 914.2 The exponential form of Fourier series 964.3 General intervals 984.4 Parseval’s theorem 1034.5 Back to the delta function 1054.6 Fourier transform 1074.7 Convolution integral 111Summary 115Exercises 1165 Series solutions of ordinary differential equations 1215.1 The Frobenius method 1225.2 Wronskian method for obtaining a second solution 1375.3 Bessel and Neumann functions 1375.4 Legendre polynomials 142Summary 144Exercises 1456 Spherical harmonics 1476.1 Properties of the Legendre polynomials, Pl(x) 1486.2 Associated Legendre functions, Pm l (x) 1576.3 Spherical harmonic functions, Yml (θ, φ) 1586.4 Addition theorem for Ym l (θ, φ) 1606.5 Laplace equation in spherical coordinates 166Summary 167Exercises 1687 Bessel functions 1737.1 Small-argument and asymptotic forms 1737.2 Properties of the Bessel functions, Jn(x) 1757.3 Orthogonality 1807.4 Bessel series 1827.5 The Fourier-Bessel transform 1857.6 Spherical Bessel functions 1867.7 Expansion of plane waves in spherical harmonics 190Summary 192Exercises 1928 Complex analysis 1958.1 Complex functions 1958.2 Analytic functions: differentiable in a region 1978.3 Contour integrals 2028.4 Integrating analytic functions 2068.5 Cauchy integral formulas 2108.6 Taylor and Laurent series 2138.7 Singularities and residues 2178.8 Definite integrals 2218.9 Meromorphic functions 2288.10 Approximation of integrals 2308.11 The analytic signal 2368.12 The Laplace transform 242Summary 245Exercises 2459 Inhomogeneous differential equations 2519.1 The method of Green functions 2519.2 Poisson equation 2609.3 Helmholtz equation 2669.4 Diffusion equation 2729.5 Wave equation 2799.6 The Kirchhoff integral theorem 283Summary 284Exercises 28410 Integral equations 28710.1 Introduction 28710.2 Classification of integral equations 29010.3 Neumann series 29110.4 Integral transform methods 29310.5 Separable kernels 29510.6 Self-adjoint kernels 29710.7 Numerical approaches 302Summary 314Exercises 31511 Tensor analysis 31911.1 Once over lightly: A quick intro to tensors 31911.2 Transformation properties 32711.3 Contraction and the quotient theorem 34011.4 The metric tensor 34211.5 Raising and lowering indices 34411.6 Geometric properties of covariant vectors 34711.7 Relative tensors 35011.8 Tensors as operators 35311.9 Symmetric and antisymmetric tensors 35611.10 The Levi-Civita tensor 35711.11 Pseudotensors 36011.12 Covariant differentiation of tensors 363Summary 373Exercises 374A Vector calculus 377A.1 Scalar fields 377A.1.1 The directional derivative 377A.1.2 The gradient 378A.2 Vector fields 379A.2.1 Divergence 379A.2.2 Curl 380A.2.3 The Laplacian 380A.2.4 Vector operator formulae 381A.3 Integration 382A.3.1 Line integrals 382A.3.2 Surface integrals 383A.4 Important integral theorems in vector calculus 384A.4.1 Green’s theorem in the plane 384A.4.2 The divergence theorem 386A.4.3 Stokes’ theorem 386A.4.4 Conservative fields 387A.4.5 The Helmholtz theorem 389A.5 Coordinate systems 390A.5.1 Orthogonal curvilinear coordinates 390A.5.2 Unit vectors 391A.5.3 Differential displacement 392A.5.4 Differential surface and volume elements 393A.5.5 Transformation of vector components 393A.5.6 Cylindrical coordinates 394B Power series 401C The gamma function, Γ(x) 403Recursion relation 403Limit formula 404Reflection formula 405Digamma function 405D Boundary conditions for Partial Differential Equations 409Summary 417References 419Index 421

BRETT BORDEN, PHD, joined the Faculty of the Naval Postgraduate School in Monterey, CA in 2002, where he is Emeritus Professor of Physics. He served on the editorial board of the journal Inverse Problems from 2002 to 2013. Dr. Borden is a Fellow of The Institute of Physics. JAMES LUSCOMBE, PHD, is a Professor at the Naval Postgraduate School in Monterey, California. He teaches a variety of topics, including general relativity, statistical mechanics, mathematical methods, and quantum computation.

• Preface xi1 Vectors and linear operators 11.1 The linearity of physical phenomena 11.2 Vector spaces 21.3 Inner products and orthogonality 101.4 Operators and matrices 161.5 Eigenvectors and their role in representing operators 361.6 Hilbert space: Infinite-dimensional vector space 43Exercises 472 Sturm–Liouville theory 512.1 Second-order differential equations 522.2 Sturm–Liouville systems 572.3 The Sturm–Liouville eigenproblem 602.4 The Dirac delta function 642.5 Completeness 662.6 Recap 68Summary 68Exercises 693 Partial differential equations 713.1 A survey of partial differential equations 713.2 Separation of variables and the Helmholtz equation 763.3 The paraxial approximation 833.4 The three types of linear PDEs 843.5 Outlook 88Summary 88Exercises 894 Fourier analysis 914.1 Fourier series 914.2 The exponential form of Fourier series 964.3 General intervals 984.4 Parseval’s theorem 1034.5 Back to the delta function 1054.6 Fourier transform 1074.7 Convolution integral 111Summary 115Exercises 1165 Series solutions of ordinary differential equations 1215.1 The Frobenius method 1225.2 Wronskian method for obtaining a second solution 1375.3 Bessel and Neumann functions 1375.4 Legendre polynomials 142Summary 144Exercises 1456 Spherical harmonics 1476.1 Properties of the Legendre polynomials, Pl(x) 1486.2 Associated Legendre functions, Pm l (x) 1576.3 Spherical harmonic functions, Yml (θ, φ) 1586.4 Addition theorem for Ym l (θ, φ) 1606.5 Laplace equation in spherical coordinates 166Summary 167Exercises 1687 Bessel functions 1737.1 Small-argument and asymptotic forms 1737.2 Properties of the Bessel functions, Jn(x) 1757.3 Orthogonality 1807.4 Bessel series 1827.5 The Fourier-Bessel transform 1857.6 Spherical Bessel functions 1867.7 Expansion of plane waves in spherical harmonics 190Summary 192Exercises 1928 Complex analysis 1958.1 Complex functions 1958.2 Analytic functions: differentiable in a region 1978.3 Contour integrals 2028.4 Integrating analytic functions 2068.5 Cauchy integral formulas 2108.6 Taylor and Laurent series 2138.7 Singularities and residues 2178.8 Definite integrals 2218.9 Meromorphic functions 2288.10 Approximation of integrals 2308.11 The analytic signal 2368.12 The Laplace transform 242Summary 245Exercises 2459 Inhomogeneous differential equations 2519.1 The method of Green functions 2519.2 Poisson equation 2609.3 Helmholtz equation 2669.4 Diffusion equation 2729.5 Wave equation 2799.6 The Kirchhoff integral theorem 283Summary 284Exercises 28410 Integral equations 28710.1 Introduction 28710.2 Classification of integral equations 29010.3 Neumann series 29110.4 Integral transform methods 29310.5 Separable kernels 29510.6 Self-adjoint kernels 29710.7 Numerical approaches 302Summary 314Exercises 31511 Tensor analysis 31911.1 Once over lightly: A quick intro to tensors 31911.2 Transformation properties 32711.3 Contraction and the quotient theorem 34011.4 The metric tensor 34211.5 Raising and lowering indices 34411.6 Geometric properties of covariant vectors 34711.7 Relative tensors 35011.8 Tensors as operators 35311.9 Symmetric and antisymmetric tensors 35611.10 The Levi-Civita tensor 35711.11 Pseudotensors 36011.12 Covariant differentiation of tensors 363Summary 373Exercises 374A Vector calculus 377A.1 Scalar fields 377A.1.1 The directional derivative 377A.1.2 The gradient 378A.2 Vector fields 379A.2.1 Divergence 379A.2.2 Curl 380A.2.3 The Laplacian 380A.2.4 Vector operator formulae 381A.3 Integration 382A.3.1 Line integrals 382A.3.2 Surface integrals 383A.4 Important integral theorems in vector calculus 384A.4.1 Green’s theorem in the plane 384A.4.2 The divergence theorem 386A.4.3 Stokes’ theorem 386A.4.4 Conservative fields 387A.4.5 The Helmholtz theorem 389A.5 Coordinate systems 390A.5.1 Orthogonal curvilinear coordinates 390A.5.2 Unit vectors 391A.5.3 Differential displacement 392A.5.4 Differential surface and volume elements 393A.5.5 Transformation of vector components 393A.5.6 Cylindrical coordinates 394B Power series 401C The gamma function, Γ(x) 403Recursion relation 403Limit formula 404Reflection formula 405Digamma function 405D Boundary conditions for Partial Differential Equations 409Summary 417References 419Index 421