Mathematical methods in engineering and physics : special functions and boundary-value problems / David E. Johnson, Johnny R. Johnson.
Editor: New York : Ronald Press, c1965Descripción: viii, 273 p. ; 24 cmTema(s): Engineering mathematics | Mathematical physicsOtra clasificación: 00A061 ORTHOGONAL FUNCTIONS I
1.1 Introduction, [1]
1.2 Preliminary Definitions, [4]
1.3 Generalized Fourier Series, [6]
1.4 Bessel’s Inequality, [7]
1.5 Parseval’s Equation, [7]
1.6 An Orthogonalization Process, [9]
1.7 An Example, [11]
1.8 Generating Functions and Recurrence Relations, [13]
1.9 Sturm-Liouville Systems, [17]
1.10 Orthogonal Functions of Two Variables, [20]
1.11 Rodrigues’ Formula, [22]
2 FOURIER SERIES [25]
2.1 Fourier Trigonometric Series, [25]
2.2 An Example, [27]
2.3 Even and Odd Functions, [29]
2.4 Expansions of Even and Odd Functions; Half-Range Series, [30]
2.5 Operations with Fourier Series, [34]
2.6 Parseval’s Theorem, [38]
2.7 Fourier Integral, [39]
3 SERIES SOLUTION OF DIFFERENTIAL EQUATIONS [42]
3.1 Preliminary Example, [42]
3.2 Generalized Series Solution, [45]
3.3 Existence of the Series Solution, [46]
3.4 A Second Solution, [48]
4 LEGENDRE FUNCTIONS [51]
4.1 Legendre Polynomials, [51]
4.2 Legendre Functions of the Second Kind, [54]
4.3 Generating Function for Pn(x), [55]
4.4 Rodrigues’ Formula, [57]
4.5 Orthogonality of the Pn(x), [58]
4.6 Recurrence Relations for Pn(x), [59]
4.7 Series Expansions Involving Pn(x), [61]
4.8 Associated Legendre Functions, [64]
4.9 Recurrence Relations for Pnm(x), [65]
4.10 Orthogonality and Generating Function of Pnm(x), QI
4.11 Spherical Harmonics, [70]
4.12 Series Expansions Involving Pnm(x)t [72]
4.13 Another Expression for Qn(x), [74]
4.14 Recurrence Relations for Qn(x), [15]
4.15 Generating Function for Qn(x), [77]
5 THE GAMMA FUNCTION [79]
5.1 Integral Definition, [79]
5.2 Euler’s Constant, [81]
5.3 Weierstrass’ Definition, [83]
5.4 Other Forms for the Gamma Function, [84]
5.5 Logarithmic Derivative, [86]
6 BESSEL FUNCTIONS [88]
6.1 Bessel’s Differential Equation, [88]
6.2 Bessel Function of the Second Kind, [91]
6.3 Generating Function for Jn(x), [94]
6.4 Recurrence Relations, [97]
6.5 Spherical Bessel Functions, [99]
6.6 Zeros of Jn(x), [101]
6.7 Orthogonality of Jn(x), [102]
6.8 Integral Relations, [105]
6.9 Some Properties of Yn(x), [109]
6.10 An Orthogonality Relation Involving Yn(x), [111]
7 BOUNDARY-VALUE PROBLEMS [113]
7.1 Linear Operators and Boundary-Value Problems, [113]
7.2 Principle of Superposition, [115]
7.3 Infinite Series of Solutions, [116]
7.4 Separation-of-Variables Method, [119]
7.5 Summary of the Method, [120]
7.6 An Example, [122]
7.7 Limitations of the Method, [125]
8 PARTIAL DIFFERENTIAL EQUATIONS OF MATHEMATICAL PHYSICS [129]
8.1 Helmholtz Equation, [129]
8.2 Wave Equation, [131]
8.3 Vibrating String, [131]
8.4 Vibrating Membrane, [134]
8.5 Diffusion Equation, [136]
8.6 Laplace’s Equation, [140]
9 HERMITE POLYNOMIALS [145]
9.1 Definition, [145]
9.2 Generating Function, [146]
9.3 Recurrence Relations, [147]
9.4 Orthogonality, [148]
9.5 Expansion of Functions in Terms of Hn(x), [149]
9.6 General Solution of Hermite’s Equation, [151]
9.7 Hermite’s Orthogonal Functions, [152]
10 LAGUERRE POLYNOMIALS [154]
10.1 Definition, [154]
10.2 Recurrence Relations and Differential Equation, [155]
10.3 Rodrigues’ Formula, [158]
10.4 Orthogonality, [158]
10.5 Simple Laguerre Polynomials Ln(x), [160]
10.6 Example from Quantum Mechanics, [163]
11 CHEBYSHEV POLYNOMIALS [166]
11.1 Definitions, [166]
11.2 Recurrence Relations and Differential Equations, [167]
11.3 Orthogonality Relations, [169]
11.4 Generating Functions, [171]
11.5 Rodrigues’ Formula, [173]
11.6 Zeros of Tn(x) and Associated Properties, [174]
11.7 Expansions in Series of Chebyshev Polynomials, [175]
11.8 An Approximation Example, [179]
11.9 Boundary-Value Problems, [180]
12 MATHIEU FUNCTIONS [183]
12.1 Mathieu’s Equation, [183]
12.2 Properties of Elliptic-Cylinder Coordinates, [184]
12.3 Solution of Mathieu’s Equation, [185]
12.4 Nature of the General Solutions, [189]
12.5 Orthogonality of the Periodic Solutions, [191]
12.6 An Example, [193]
13 OTHER SPECIAL FUNCTIONS [196]
13.1 Hypergeometric Function, [196]
13.2 Jacobi Polynomials, [200]
13.3 Rodrigues’ Formula for Jacobi Polynomials, [201]
13.4 Orthogonality of the Jacobi Polynomials, [202]
13.5 Bessel Polynomials, [205]
13.6 Some Related Polynomials, [206]
14 LAPLACE AND FOURIER TRANSFORMS [210]
14.1 Introduction, [210]
14.2 Laplace Transform, [211]
14.3 Solutions of Differential Equations, [214]
14.4 Convolution, [216]
14.5 Fourier Transform, [219]
14.6 Properties of the Fourier Transform, [221]
14.7 System Functions, [223]
14.8 Filter Theory, [224]
15 STURM-LIOUVILLE TRANSFORMS [229]
15.1 Definition, [229]
15.2 Finite Fourier Sine and Cosine Transforms, [230]
15.3 Hankel Transform, [234]
15.4 Legendre Transform, [236]
15.5 Laguerre Transform, [238]
15.6 Hermite Transform, [239]
15.7 Other Transforms, [239]
16 A GENERAL CLASS OF ORTHOGONAL POLYNOMIALS [242]
16.1 A Unifying Concept, [242]
16.2 Orthogonality of Gn, [244]
16.3 Norm of Gn{ -1, α + β, -αβ, h, k, Cn, x), [245]
16.4 Infinite Intervals, [248]
16.5 Generating Functions, [249]
16.6 Summary, [252]
APPENDIX
A PROPERTIES OF INFINITE SERIES [255]
A.l Convergent Series, [255]
A.2 Uniformly Convergent Series, [256]
A. 3 Power Series, [258]
B CONVERGENCE OF THE FOURIER SERIES [259]
B. l Sufficiency for Convergence, [259]
C TABLES [262]
1 Laplace Transforms, [262]
2 Finite Sine Transforms, [264]
3 Finite Cosine Transforms, [264]
4 Summary of Properties of Polynomial Sets {ϕn(x)}> [265]
5 Generating Functions, [266]| Item type | Home library | Shelving location | Call number | Materials specified | Status | Date due | Barcode |
|---|---|---|---|---|---|---|---|
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| 00A06 F646 Mathematische Methoden der Physik / | 00A06 G798-2 Advanced engineering mathematics / | 00A06 H642 Advanced calculus for applications / | 00A06 J66 Mathematical methods in engineering and physics : | 00A06 J82 Mathematical techniques : | 00A06 K92-7m Maple computer manual : | 00A06 K92-8m Maple computer guide : |
Bibliografía: p. 267-268.
1 ORTHOGONAL FUNCTIONS I --
1.1 Introduction, [1] --
1.2 Preliminary Definitions, [4] --
1.3 Generalized Fourier Series, [6] --
1.4 Bessel’s Inequality, [7] --
1.5 Parseval’s Equation, [7] --
1.6 An Orthogonalization Process, [9] --
1.7 An Example, [11] --
1.8 Generating Functions and Recurrence Relations, [13] --
1.9 Sturm-Liouville Systems, [17] --
1.10 Orthogonal Functions of Two Variables, [20] --
1.11 Rodrigues’ Formula, [22] --
2 FOURIER SERIES [25] --
2.1 Fourier Trigonometric Series, [25] --
2.2 An Example, [27] --
2.3 Even and Odd Functions, [29] --
2.4 Expansions of Even and Odd Functions; Half-Range Series, [30] --
2.5 Operations with Fourier Series, [34] --
2.6 Parseval’s Theorem, [38] --
2.7 Fourier Integral, [39] --
3 SERIES SOLUTION OF DIFFERENTIAL EQUATIONS [42] --
3.1 Preliminary Example, [42] --
3.2 Generalized Series Solution, [45] --
3.3 Existence of the Series Solution, [46] --
3.4 A Second Solution, [48] --
4 LEGENDRE FUNCTIONS [51] --
4.1 Legendre Polynomials, [51] --
4.2 Legendre Functions of the Second Kind, [54] --
4.3 Generating Function for Pn(x), [55] --
4.4 Rodrigues’ Formula, [57] --
4.5 Orthogonality of the Pn(x), [58] --
4.6 Recurrence Relations for Pn(x), [59] --
4.7 Series Expansions Involving Pn(x), [61] --
4.8 Associated Legendre Functions, [64] --
4.9 Recurrence Relations for Pnm(x), [65] --
4.10 Orthogonality and Generating Function of Pnm(x), QI --
4.11 Spherical Harmonics, [70] --
4.12 Series Expansions Involving Pnm(x)t [72] --
4.13 Another Expression for Qn(x), [74] --
4.14 Recurrence Relations for Qn(x), [15] --
4.15 Generating Function for Qn(x), [77] --
5 THE GAMMA FUNCTION [79] --
5.1 Integral Definition, [79] --
5.2 Euler’s Constant, [81] --
5.3 Weierstrass’ Definition, [83] --
5.4 Other Forms for the Gamma Function, [84] --
5.5 Logarithmic Derivative, [86] --
6 BESSEL FUNCTIONS [88] --
6.1 Bessel’s Differential Equation, [88] --
6.2 Bessel Function of the Second Kind, [91] --
6.3 Generating Function for Jn(x), [94] --
6.4 Recurrence Relations, [97] --
6.5 Spherical Bessel Functions, [99] --
6.6 Zeros of Jn(x), [101] --
6.7 Orthogonality of Jn(x), [102] --
6.8 Integral Relations, [105] --
6.9 Some Properties of Yn(x), [109] --
6.10 An Orthogonality Relation Involving Yn(x), [111] --
7 BOUNDARY-VALUE PROBLEMS [113] --
7.1 Linear Operators and Boundary-Value Problems, [113] --
7.2 Principle of Superposition, [115] --
7.3 Infinite Series of Solutions, [116] --
7.4 Separation-of-Variables Method, [119] --
7.5 Summary of the Method, [120] --
7.6 An Example, [122] --
7.7 Limitations of the Method, [125] --
8 PARTIAL DIFFERENTIAL EQUATIONS OF MATHEMATICAL PHYSICS [129] --
8.1 Helmholtz Equation, [129] --
8.2 Wave Equation, [131] --
8.3 Vibrating String, [131] --
8.4 Vibrating Membrane, [134] --
8.5 Diffusion Equation, [136] --
8.6 Laplace’s Equation, [140] --
9 HERMITE POLYNOMIALS [145] --
9.1 Definition, [145] --
9.2 Generating Function, [146] --
9.3 Recurrence Relations, [147] --
9.4 Orthogonality, [148] --
9.5 Expansion of Functions in Terms of Hn(x), [149] --
9.6 General Solution of Hermite’s Equation, [151] --
9.7 Hermite’s Orthogonal Functions, [152] --
10 LAGUERRE POLYNOMIALS [154] --
10.1 Definition, [154] --
10.2 Recurrence Relations and Differential Equation, [155] --
10.3 Rodrigues’ Formula, [158] --
10.4 Orthogonality, [158] --
10.5 Simple Laguerre Polynomials Ln(x), [160] --
10.6 Example from Quantum Mechanics, [163] --
11 CHEBYSHEV POLYNOMIALS [166] --
11.1 Definitions, [166] --
11.2 Recurrence Relations and Differential Equations, [167] --
11.3 Orthogonality Relations, [169] --
11.4 Generating Functions, [171] --
11.5 Rodrigues’ Formula, [173] --
11.6 Zeros of Tn(x) and Associated Properties, [174] --
11.7 Expansions in Series of Chebyshev Polynomials, [175] --
11.8 An Approximation Example, [179] --
11.9 Boundary-Value Problems, [180] --
12 MATHIEU FUNCTIONS [183] --
12.1 Mathieu’s Equation, [183] --
12.2 Properties of Elliptic-Cylinder Coordinates, [184] --
12.3 Solution of Mathieu’s Equation, [185] --
12.4 Nature of the General Solutions, [189] --
12.5 Orthogonality of the Periodic Solutions, [191] --
12.6 An Example, [193] --
13 OTHER SPECIAL FUNCTIONS [196] --
13.1 Hypergeometric Function, [196] --
13.2 Jacobi Polynomials, [200] --
13.3 Rodrigues’ Formula for Jacobi Polynomials, [201] --
13.4 Orthogonality of the Jacobi Polynomials, [202] --
13.5 Bessel Polynomials, [205] --
13.6 Some Related Polynomials, [206] --
14 LAPLACE AND FOURIER TRANSFORMS [210] --
14.1 Introduction, [210] --
14.2 Laplace Transform, [211] --
14.3 Solutions of Differential Equations, [214] --
14.4 Convolution, [216] --
14.5 Fourier Transform, [219] --
14.6 Properties of the Fourier Transform, [221] --
14.7 System Functions, [223] --
14.8 Filter Theory, [224] --
15 STURM-LIOUVILLE TRANSFORMS [229] --
15.1 Definition, [229] --
15.2 Finite Fourier Sine and Cosine Transforms, [230] --
15.3 Hankel Transform, [234] --
15.4 Legendre Transform, [236] --
15.5 Laguerre Transform, [238] --
15.6 Hermite Transform, [239] --
15.7 Other Transforms, [239] --
16 A GENERAL CLASS OF ORTHOGONAL POLYNOMIALS [242] --
16.1 A Unifying Concept, [242] --
16.2 Orthogonality of Gn, [244] --
16.3 Norm of Gn{ -1, α + β, -αβ, h, k, Cn, x), [245] --
16.4 Infinite Intervals, [248] --
16.5 Generating Functions, [249] --
16.6 Summary, [252] --
APPENDIX --
A PROPERTIES OF INFINITE SERIES [255] --
A.l Convergent Series, [255] --
A.2 Uniformly Convergent Series, [256] --
A. 3 Power Series, [258] --
B CONVERGENCE OF THE FOURIER SERIES [259] --
B. l Sufficiency for Convergence, [259] --
C TABLES [262] --
1 Laplace Transforms, [262] --
2 Finite Sine Transforms, [264] --
3 Finite Cosine Transforms, [264] --
4 Summary of Properties of Polynomial Sets {ϕn(x)}> [265] --
5 Generating Functions, [266] --
MR, 33 #1992
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