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| Àâòîðèçàöèÿ |
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| Ïîèñê ïî óêàçàòåëÿì |
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| Hanna J.R., Rowland J.H. — Fourier Series, Transforms, and Boundary Value Problems |
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| Ïðåäìåòíûé óêàçàòåëü |
Abbreviations,  (set of natural numbers plus zero) 45
Abbreviations, Abel’s uniform convergence test 243
Abbreviations, AI (absolutely integrable) 107
Abbreviations, BVP* (boundary value problem) 1
Abbreviations, DFT* (discrete Fourier transform) 198
Abbreviations, FFT* (fast Fourier transform) 203
Abbreviations, IVP* (initial value problem) 2
Abbreviations, M (any set of n consecutive integers) 199
Abbreviations, N (set of natural numbers) 43
Abbreviations, ODE* (ordinary differential equation) 1
Abbreviations, PDE* (partial differential equation) 1
Abbreviations, PWC (piecewise continuous) 68
Abbreviations, PWS (piecewise smooth) 70
Abbreviations, SI (square integrable) 64
Abbreviations, SLDE* (Sturm — Liouville differential equation) 50
Abbreviations, SLP* (Sturm — Liouville problem) 50
Abbreviations, Z (set of all integers) 48
Absolutely integrable (AI) 107
Adjoint of operator 58
Airy’s equation 18
Approximation in mean 64 198—202
Arbitrary constants 1 6 11—12
Arbitrary functions 23 26 33
Associated Legendre differential equation 162
Associated Legendre functions 164
Associated Legendre polynomials 163
Auxiliary equation 5
Basic Fourier Series 72—74
bernoulli 68
Bessel equation 16 117
Bessel equation, modified type 121
Bessel equation, Sturm — Liouville type 134—135
Bessel functions 18 117 119
Bessel functions, boundedness of 132
Bessel functions, derivatives of 124—125
Bessel functions, first kind 18 119
Bessel functions, generating functions for 130
Bessel functions, graphs of 122
Bessel functions, integral form for 130—132
Bessel functions, integrals of 127—129
Bessel functions, modified 121
Bessel functions, norms 137
Bessel functions, orthogonal sets of 135 137
Bessel functions, second kind 121
Bessel functions, series of 137—139
Bessel functions, zeros of 134—137
Bessel transforms 210
Bessel’s inequality 65
boundary conditions 2 24 219
Boundary conditions, Churchill type 248
Boundary conditions, Dirichlet type 247
Boundary conditions, linear homogeneous 219 222—225
Boundary conditions, Neumann type 248
Boundary conditions, nonhomogeneous 219 225
Boundary conditions, Robin type 248
Boundary value problems (BVPs) 1 24 219 270
Boundary value problems (BVPs), boundedness of solutions 36 222
Boundary value problems (BVPs), defined 24
Boundary value problems (BVPs), methods for solving 173 186 219 222—225 270—273 283—285 307-310
Boundary value problems (BVPs), solutions verified 225—228 242—245
Cauchy equation 7
Cauchy principal value 113—114
Cauchy — Kowalewsky theorem 25
Characteristic equations 5 8 27—31
Characteristic functions 50
Characteristic values 50
Classification, second order PDEs 23—24
Complete orthogonal sets 65—66
Complex conjugate 46—47 198
Complex-valued functions 46—47
Conduction of heat 236—238
Conductivity, thermal 236
Continuous functions 45 59
Continuous functions, piecewise (PWC) 68
Continuous functions, sectionally 68
Convergence in mean 64 66
Convergence of Fourier cosine series 78—79
Convergence of Fourier integrals 108
Convergence of Fourier series 75—76
Convergence of Fourier sine series 79
Convergence, point wise 66
Convergence, point wise, uniform, integrals 102—103
Convergence, point wise, uniform, series 58—59
Convolution 180—181
Convolution for Fourier exponential transforms 192—193
Convolution for Laplace transforms 180—181
Convolution for Mellin transforms 217
Cylindrical coordinates 140
Cylindrical coordinates, Laplacian in 141
Derivative, left hand 69
Derivative, right hand 69
Differential equations, Cauchy (Euler) 7
Differential equations, linear 2
Differential equations, linear homogeneous 2
Differential equations, nonhomogeneous 6
Differential equations, ODEs 1
Differential equations, PDEs 1
Differential equations, systems of 185
Differential operator 2
Differentiation of series 9 92—94
Diffusion, coefficient of 238
Diffusion, equation 238
Dirac delta function 177—179
Dirichlet problem 247—248
Discrete Fourier Transform 197—203
Discrete inner product 198
Discrete least squares approximation 198
Discrete norm 198
D’Alembert 68
d’Alembert solution of 28 34
d’Alembert’s solution 28 34
Eigenfunctions 50
Eigenfunctions, linear independence of 57
Eigenfunctions, series of 61 62
Eigenvalues 50
Elasticity, modulus 230
electrical circuits 271 273
Elliptic type, PDEs of 23—24
Error function 183 301—305
Euler 7 68
Euler, constant 120—121
Euler, differential equation 7
Euler, formulas 74
Euler, identity (relation) 32 112
Even function 76—77
Existence 2 25 169—170
Exponential, Fourier integrals 112—114
Exponential, Fourier series 80—82
Exponential, Fourier transforms 190 192
Exponential, function 5—6 30—31 46-48
Exponential, order 169
Exponential, solutions 5—6 30—31
Extension, nonperiodic 111
Extension, periodic 74 78
Fast Fourier Transform 203—205
Finite difference formulas 200
Finite difference method 241—242
Forward difference method 242
FOURIER 68
Fourier method of solution 222—225
Fourier method of solution, two dimensional 276—280
Fourier series 72—74
Fourier series in two variables 97—99
Fourier series, convergence 75—79
Fourier series, cosine 78
Fourier series, differentiation 92—94
Fourier series, exponential form 80—81
Fourier series, generalized 62
Fourier series, integration 94—96
Fourier series, sine 78—79
| Fourier series, uniform convergence of 89—92
Fourier sine integral formula 110
Fourier sine integral formula, theorem 75—76
Fourier sine integral formula, transforms of two variables 208—209
Fourier sine integral formula, transforms, cosine kernel 191
Fourier sine integral formula, transforms, exponential kernel 192
Fourier sine integral formula, transforms, sine kernel 191
Fourier — Bessel series 137—139
Fourier, constants (coefficients) 62—65 73—74 78—79 81
Fourier, cosine form 109—110
Fourier, cosine kernel for 189
Fourier, exponential form 112—113
Fourier, exponential kernel for 190
Fourier, finite transforms 189—190
Fourier, integral formula 107—108
Fourier, integral theorem 108 110—111
Fourier, method 219 270
Fourier, sine form 110
Fourier, sine kernel for 189
Functions, analytic 10 25
Functions, associated 164
Functions, Bessel 18 119—121
Functions, characteristic 50
Functions, complex-valued 46—47
Functions, continuous 45—46 59
Functions, error 183 301—305
Functions, even 76—77
Functions, exponential 5—6 30—31 46-48
Functions, gamma 117—119 170
Functions, generating for Bessel functions 130—132
Functions, generating for Legendre polynomials 149—151
Functions, harmonic 161 248
Functions, hyperbolic 47
Functions, inner product of 43 198
Functions, Legendre 144
Functions, normalized 43
Functions, norms of 43 198
Functions, odd 76—77
Functions, orthogonal 43
Functions, orthonormal 43
Functions, piecewise continuous 68—70
Functions, piecewise smooth 70
Fundamental interval 74
Fundamental set 4
Gamma function 117—119 170
Gauss — Seidel method 251—252
General solution 4 6 28—33
Generalized Fourier series 62
Generalized functions (or distributions) 179
Geometric series (progressions) 180 199
Gibbs phenomenon 86—88
Gram — Schmidt orthogonalization 44—45
Hadamard example 25—26
Hankel transforms 209—212
Harmonic analysis 82—84 198—202
Harmonic functions 161 248
Harmonic functions in cylindrical regions 260—263
Harmonic functions in half-plane 312
Harmonic functions in rectangular regions 248—251
Harmonic functions in spherical regions 267—268
Harmonic functions in strip 313—314
Heat conduction, equation 238
Heat conduction, experimental observations 236
Heat conduction, flow 236
Heat conduction, problem 236—238
Heat conduction, problem, solution 238—241
Heat conduction, problem, uniqueness 244—245
Heat conduction, problem, verification 242—244
Hermite polynomials 49
Hermitian orthogonality 47—49
Homogeneous equations 2 5—7 23
Hooke’s law 270
Hyperbolic functions 47
Hyperbolic type, PDEs of 23—24
Improper integral, principal value of 113—114
Improper integral, uniform convergence of 102—103
Indicial equation 14
Indicial equation, roots of 14
Inequality, Bessel’s 65
Inequality, Schwarz 46
Initial value problems (IVPs) 2—3 24
Inner product 40 43 198
inner product of functions 43
Inner product of vectors 40
Insulated surface 238 245 254 291
Integral equation 184
Integral form, Bessel’s function 130—132
Integral theorem, Fourier 108
Integral transforms 168 189 191 208—210 214—215
Integral transforms, Fourier 191
Integral transforms, Fourier, cosine 191
Integral transforms, Fourier, exponential 192
Integral transforms, Fourier, finite 189—190
Integral transforms, Fourier, finite, cosine 189
Integral transforms, Fourier, finite, exponential 190
Integral transforms, Fourier, finite, sine 189
Integral transforms, Fourier, sine 191
Integral transforms, Fourier, two variables 208
Integral transforms, Hankel 209—210
Integral transforms, Hankel, finite 210
Integral transforms, kernels of 210
Integral transforms, Laplace 168
Integral transforms, Legendre 214
Integral transforms, Mellin 215
Integrating factor 50
Integration of series 94—96
Integro-differential equations 184
Interpolation 202
Inverse transforms 173 189—192 198 208—211 214 217
Jump discontinuity 69
Kernel 168
Kirchhoff s Law 271
Lagrange’s relation 58
Laguerre polynomials 49
Laplace transforms 168
Laplace’s equation 247
Laplacian in cylindrical coordinates 141
Laplacian in rectangular coordinates 247
Laplacian in spherical coordinates 160—161 164
Least squares approximation 64—66
Left hand, derivative 69—70
Left hand, limit 68—69
Legendre, equation 12 142
Legendre, functions of second kind 164
Legendre, functions, associated 164
Legendre, polynomials 142 144
Legendre, polynomials, associated 162—163
Legendre, polynomials, derivatives of 148 151
Legendre, polynomials, generating functions 149
Legendre, polynomials, norms 153—154
Legendre, polynomials, orthogonal sets of 152—153
Legendre, polynomials, Rodrigues’ formula 146—148
Legendre, polynomials, series of 154—155
Legendre, transforms 214—215
Liebmann method 251—252
Limits in the mean 64 66
Limits, one sided 68
Linear combinations 2 4
Linear combinations, extensions of, by integrals 111 297
Linear combinations, extensions of, by series 74 78 224—225 229 240 250
Linear dependence (independence) 3
Linear dependence (independence) of eigenfunctions 57
Linear dependence (independence) of functions 3
Linear differential equations 2 4—6 23
Linear operators 1—2
Linear operators, adjoint of 58
Linear operators, product of 2
Linear operators, self-adjoint 58
Linear operators, sum of 1
Liouville (Sturm — Liouville problem) 50
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