| Start page | 0
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| Start page using frames | 0
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| Electronic release notes | 0
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| Page index (hyperlinked) | 0
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| Subject index (hyperlinked) | 0
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| Title page | I (HTML)
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| Errata Notice | II (HTML)
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| Preface | III (HTML)
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| Preface to the Ninth Printing | IIIa (HTML)
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| Foreword | V (HTML)
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| | VI
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| Table of Contents | VII (HTML)
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| | VIII
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| Introduction. 1. Introduction. 2. Accuracy of the Tables. | IX (HTML)
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| 3. Auxiliary Functions and Arguments. 4. Interpolation | X
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| | XI
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| 5. Inverse Interpolation | XII
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| 6. Bivariate Interpolation. 7. Generation of Functions from Recurrence Relations | XIII
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| 8. Acknowledgments | XIV (HTML)
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| 2. Physical Constants and Conversion Factors | 5
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| Table 2.1. Common Units and Conversion Factors. Table 2.2. Names and Conversion Factors for Electric and Magnetic Units | 6
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| Table 2.3. Adjusted Values of Constants | 7
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| Table 2.4. Miscellaneous Conversion Factors. Table 2.5. Conversion Factors for Customary U.S. Units to Metric Units. Table 2.6. Geodetic Constants | 8
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| 3. Elementary analytical methods | 9
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| 3.1. Binomial Theorem and Binomial Coefficients; Arithmetic and Geometric Progressions; Arithmetic, Geometric, Harmonic and Generalized Means. 3.2. Inequalities | 10
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| 3.3. Rules for Differentiation and Integration | 11
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| | 12
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| 3.4. Limits, Maxima and Minima | 13
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| 3.5. Absolute and Relative Errors. 3.6. Infinite Series | 14
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| | 15
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| 3.7. Complex Numbers and Functions | 16
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| 3.8. Algebraic Equations | 17
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| 3.9. Successive Approximation Methods | 18
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| 3.10. Theorems on Continued Fractions. Numerical Methods. 3.11. Use and Extension of the Tables. 3.12. Computing Techniques | 19
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| | 20
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| References | 23
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| 4. Elementary Transcendental Functions: Logarithmic, Exponential, Circular and Hyperbolic Functions | 65
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| Mathematical Properties. 4.1. Logarithmic Function | 67
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| | 68
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| 4.2. Exponential Function | 69
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| | 70
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| 4.3. Circular Functions | 71
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| | 72
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| | 73
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| | 74
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| | 75
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| | 76
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| | 77
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| | 78
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| 4.4. Inverse Circular Functions | 79
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| | 80
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| | 81
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| | 82
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| 4.5. Hyperbolic Functions | 83
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| | 84
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| | 85
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| 4.6. Inverse Hyperbolic Functions | 86
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| | 87
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| | 88
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| Numerical Methods. 4.7. Use and Extension of the Tables | 89
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| References | 93
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| | 94
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| 5. Exponential Integral and Related Functions | 227
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| Mathematical Properties. 5.1. Exponential Integral | 228
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| | 229
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| | 230
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| 5.2. Sine and Cosine Integrals | 231
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| | 232
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| Numerical Methods. 5.3. Use and Extension of the Tables | 233
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| | 234
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| References | 235
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| | 236
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| | 237
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| 6. Gamma Function and Related Functions | 253
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| Mathematical Properties. 6.1. Gamma Function | 255
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| | 256
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| | 257
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| 6.2. Beta Function. 6.3. Psi (Digamma) Function | 258
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| | 259
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| 6.4. Polygamma Functions. 6.5. Incomplete Gamma Function | 260
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| | 261
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| | 262
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| 6.6. Incomplete Beta Function. Numerical Methods. 6.7. Use and Extension of the Tables | 263
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| 6.8. Summation of Rational Series by Means of Polygamma Functions | 264
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| References | 265
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| | 266
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| 7. Error Function and Fresnel Integrals | 295
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| Mathematical Properties. 7.1. Error Function | 297
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| | 298
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| 7.2. Repeated Integrals of the Error Function | 299
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| 7.3. Fresnel Integrals | 300
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| | 301
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| 7.4. Definite and Indefinite Integrals | 302
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| | 303
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| Numerical Methods. 7.5. Use and Extension of the Tables | 304
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| References | 308
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| | 309
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| Complex zeros, maxima, minima of the error function and Fresnel integrals: asymptotics | 329
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| 8. Legendre function | 331
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| Mathematical Properties. Notation. 8.1. Differential Equation | 332
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| 8.2. Relations Between Legendre Functions. 8.3. Values on the Cut. 8.4. Explicit Expressions | 333
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| 8.6. Special Values | 334
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| 8.7. Trigonometric Expansions. 8.8. Integral Representations. 8.9. Summation Formulas. 8.10. Asymptotic Expansions | 335
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| 8.11. Toroidal Functions | 336
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| 8.12. Conical Functions. 8.13. Relation to Elliptic Integrals. 8.14. Integrals | 337
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| | 338
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| Numerical Methods. 8.15. Use and Extension of the Tables | 339
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| References | 340
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| | 341
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| 9. Bessel Functions of Integer Order | 355
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| Mathematical Properties. Notation. Bessel Functions J and Y. 9.1. Definitions and Elementary Properties | 358
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| | 359
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| | 360
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| | 361
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| | 362
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| | 363
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| 9.2. Asymptotic Expansions for Large Arguments | 364
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| 9.3. Asymptotic Expansions for Large Orders | 365
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| | 366
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| | 367
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| | 368
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| 9.4. Polynomial Approximations | 369
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| 9.5. Zeros | 370
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| | 371
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| | 372
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| | 373
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| Modified Bessel Functions I and K. 9.6. Definitions and Properties | 374
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| | 375
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| | 376
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| 9.7. Asymptotic Expansions | 377
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| 9.8. Polynomial Approximations | 378
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| Kelvin Functions. 9.9. Definitions and Properties | 379
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| | 380
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| 9.10. Asymptotic Expansions | 381
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| | 382
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| | 383
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| 9.11. Polynomial Approximations | 384
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| Numerical Methods. 9.12. Use and Extension of the Tables | 385
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| | 386
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| | 387
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| References | 388
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| | 389
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| 10. Bessel Functions of Fractional Order | 435
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| Mathematical Properties. 10.1. Spherical Bessel Functions | 437
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| | 438
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| | 439
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| | 440
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| | 441
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| 10.2. Modified Spherical Bessel Functions | 443
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| | 444
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| 10.3. Riccati-Bessel Functions | 445
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| 10.4. Airy Functions | 446
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| | 447
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| | 448
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| | 449
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| | 450
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| | 451
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| Numerical Methods. 10.5. Use and Extension of the Tables | 452
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| References | 455
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| | 456
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| 11. Integrals of Bessel Functions | 479
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| Mathematical Properties. 11.1. Simple Integrals of Bessel Functions | 480
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| | 481
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| 11.2. Repeated Integrals of Jn(z) and K0(z) | 482
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| 11.3. Reduction Formulas for Indefinite Integrals | 483
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| | 484
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| 11.4. Definite Integrals | 485
|
| | 486
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| | 487
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| Numerical Methods. 11.5. Use and Extension of the Tables | 488
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| | 489
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| References | 490
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| | 491
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| 12. Struve Functions and Related Functions | 495
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| Mathematical Properties. 12.1. Struve Function Hn(s) | 496
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| | 497
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| 12.2. Modified Struve Function Lnu(z). 12.3. Anger and Weber Functions | 498
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| Numerical Methods. 12.4. Use and Extension of the Tables | 499
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| References | 500
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| Explanations of numerical methods to compute Struve functions | 502
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| 13. Confluent Hypergeometric Functions | 503
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| Mathematical Properties. 13.1. Definitions of Kummer and Whittaker Functions | 504
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| 13.2. Integral Representations | 505
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| 13.3. Connections With Bessel Functions | 506
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| | 507
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| 13.5. Asymptotic Expansions and Limiting Forms | 508
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| 13.6. Special Cases | 509
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| 13.7. Zeros and Turning Values | 510
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| Numerical Methods. 13.8. Use and Extension of the Tables | 511
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| 13.10. Graphing M(a, b, x) | 513
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| References | 514
|
| | 515
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| 14. Coulomb Wave Functions | 537
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| Mathematical Properties. 14.1. Differential Equation, Series Expansions | 538
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| 14.2. Recurrence and Wronskian Relations. 14.3. Integral Representations. 14.4. Bessel Function Expansions | 539
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| 14.5. Asymptotic Expansions | 540
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| | 541
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| 14.6. Special Values and Asymptotic Behavior | 542
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| Numerical Methods. 14.7. Use and Extension of the Tables | 543
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| References | 544
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| 15. Hypergeometric Functions | 555
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| Mathematical Properties. 15.1. Gauss Series, Special Elementary Cases, Special Values of the Argument | 556
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| 15.2. Differentiation Formulas and Gauss' Relations for Contiguous Functions | 557
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| Integral Representations and Transformation Formulas | 558
|
| | 559
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| | 560
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| 15.4. Special Cases of F(a, b; c; z), Polynomials and Legendre Functions | 561
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| 15.5. The Hypergeometric Differential Equation | 562
|
| | 563
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| 15.6. Riemann's Differential Equation | 564
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| 15.7. Asymptotic Expansions. References | 565
|
| | 566
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| 16. Jacobian Elliptic Functions and Theta Functions | 567
|
| | 568
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| Mathematical Properties. 16.1. Introduction | 569
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| 16.2. Classification of the Twelve Jacobian Elliptic Functions. 16.3. Relation of the Jacobian Functions to the Copolar Trio | 570
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| 16.4. Calculation of the Jacobian Functions by Use of the Arithmetic-Geometric Mean (A.G.M.). 16.5. Special Arguments. 16.6. Jacobian Functions when m=0 or 1 | 571
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| 16.7. Principal Terms. 16.8. Change of Argument | 572
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| 16.9. Relations Between the Squares of the Functions. 16.10. Change of Parameter. 16.11. Reciprocal Parameter (Jacobi's Real Transformation). 16.12. Descending Landen Transformation (Gauss' Transformation). 16.13. Approximation in Terms of Circular Functions. 16.14. Ascending Landen Transformation | 573
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| 16.15. Approximation in Terms of Hyperbolic Functions. 16.16. Derivatives. 16.17. Addition Theorems. 16.18. Double Arguments. 16.19. Half Arguments. 16.20. Jacobi's Imaginary Transformation | 574
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| 16.21. Complex Arguments. 16.22. Leading Terms of the Series in Ascending Powers of u. 16.23. Series Expansion in Terms of the Nome q and the Argument v. 16.24. Integrals of the Twelve Jacobian Elliptic Functions | 575
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| 16.25. Notation for the Integrals of the Squares of the Twelve Jacobian Elliptic Functions. 16.26. Integrals in Terms of the Elliptic Integral of the Second Kind. 16.27. Theta Functions; Expansions in Terms of the Nome q. 16.28. Relations Between the Squares of the Theta Functions. 16.29. Logarithmic Derivatives of the Theta Functions | 576
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| 16.30. Logarithms of Theta Functions of Sum and Difference. 16.31. Jacobi's Notation for Theta Functions. 16.32. Calculation of Jacobi's Theta Function Theta(u|m) by Use of the Arithmetic-Geometric Mean. 16.33. Addition of Quarter-Periods to Jacobins Eta and Theta Functions | 577
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| 16.34. Relation of Jacobi's Zeta Function to the Theta Functions. 16.35. Calculation of Jacobi's Zeta Function Z(u|m) by Use of the Arithmetic-Geometric Mean. 16.36. Neville's Notation for Theta Functions | 578
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| 16.37. Expression as Infinite Products. 16.38. Expression as Infinite Series. Numerical Methods. 16.39. Use and Extension of the Tables | 579
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| References | 581
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| 17. Elliptic Integrals | 587
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| Mathematical Properties. 17.1. Definition of Elliptic Integrals. 17.2. Canonical Forms | 589
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| 17.3. Complete Elliptic Integrals of the First and Second Kinds | 590
|
| | 591
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| 17.4. Incomplete Elliptic Integrals of the First and Second Kinds | 592
|
| | 593
|
| | 594
|
| | 595
|
| | 596
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| 17.5. Landen's Transformation | 597
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| 17.6. The Process of the Arithmetic-Geometric Mean | 598
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| 17.7. Elliptic Integrals of the Third Kind | 599
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| Numerical Methods. 17.8. Use and Extension of the Tables | 600
|
| | 601
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| References | 606
|
| | 607
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| 18. Weierstrass Elliptic and Related Functions | 627
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| Mathematical Properties. 18.1. Definitions, Symbolism, Restrictions and Conventions | 629
|
| | 630
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| 18.2. Homogeneity Relations, Reduction Formulas and Processes | 631
|
| | 632
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| 18.3. Special Values and Relations | 633
|
| | 634
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| 18.4. Addition and Multiplication Formulas. 18.5. Series Expansions | 635
|
| | 636
|
| | 637
|
| | 638
|
| | 639
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| 18.6. Derivatives and Differential Equations | 640
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| 18.7. Integrals | 641
|
| 18.8. Conformal Mapping | 642
|
| | 643
|
| | 644
|
| | 645
|
| | 646
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| | 647
|
| | 648
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| 18.9. Relations with Complete Elliptic Integrals K and K' and Their Parameter m and with Jacobins Elliptic Functions | 649
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| 18.10. Relations with Theta Functions | 650
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| 18.11. Expressing any Elliptic Function in Terms of P and P' | 651
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| 18.13. Equianharmonic Case (g2=0, g3=1) | 652
|
| | 653
|
| | 654
|
| | 655
|
| | 656
|
| | 657
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| 18.14. Lemniscatic Case (g2=1, g3=0) | 658
|
| | 659
|
| | 660
|
| | 661
|
| 18.15. Pseudo-Lemniscatic Case (g2=-1, g3=0) | 662
|
| Numerical Methods. 18.16. Use and Extension of the Tables | 663
|
| | 664
|
| | 668
|
| | 669
|
| References | 670
|
| | 671
|
| 19. Parabolic Cylinder Functions | 685
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| Mathematical Properties. 19.1. The Parabolic Cylinder Functions, Introductory. The Equation d2y/dx2-(x2/4+a)y=0. 19.2 to 19.6. Power Series, Standard Solutions, Wronskian and Other Relations, Integral Representations, Recurrence Relations | 686
|
| | 687
|
| | 688
|
| 19.7 to 19.11. Asymptotic Expansions | 689
|
| | 690
|
| 19.12 to 19.15. Connections With Other Functions | 691
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| The Equation d2y/dx2+(x2/4-a)y=0. 19.16 to 19.19. Power Series, Standard Solutions, Wronskian and Other Relations, Integral Representations | 692
|
| 19.20 to 19.24. Asymptotic Expansions | 693
|
| | 694
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| 19.25. Connections With Other Functions | 695
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| 19.26. Zeros | 696
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| 19.27. Bessel Functions of Order ±1/4, ±3/4 as Parabolic Cylinder Functions. Numerical Methods. 19.28. Use and Extension of the Tables | 697
|
| | 698
|
| | 699
|
| References | 700
|
| 20. Mathieu Functions | 721
|
| Mathematical Properties. 20.1. Mathieu's Equation. 20.2. Determination of Characteristic Values | 722
|
| | 723
|
| | 724
|
| | 725
|
| | 726
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| 20.3. Floquet's Theorem and Its Consequences | 727
|
| | 728
|
| | 729
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| 20.4. Other Solutions of Mathieu's Equation | 730
|
| | 731
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| 20.5. Properties of Orthogonality and Normalization. 20.6. Solutions of Mathieu's Modified Equation for Integral nu | 732
|
| | 733
|
| | 734
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| 20.7. Representations by Integrals and Some Integral Equations | 735
|
| | 736
|
| | 737
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| 20.8. Other Properties | 738
|
| | 739
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| 20.9. Asymptotic Representations | 740
|
| | 741
|
| | 742
|
| | 743
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| 20.10. Comparative Notations | 744
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| References | 745
|
| | 746
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| 21. Spheroidal Wave Functions | 751
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| Mathematical Properties. 21.1. Definition of Elliptical Coordinates. 21.2. Definition of Prolate Spheroidal Coordinates. 21.3. Definition of Oblate Spheroidal Coordinates. 21.4. Laplacian in Spheroidal Coordinates. 21.5. Wave Equation in Prolate and Oblate Spheroidal Coordinates | 752
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| 21.6. Differential Equations for Radial and Angular Spheroidal Wave Functions. 21.7. Prolate Angular Functions | 753
|
| | 754
|
| | 755
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| 21.8. Oblate Angular Functions. 21.9. Radial Spheroidal Wave Functions | 756
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| 21.10. Joining Factors for Prolate Spheroidal Wave Functions | 757
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| 21.11. Notation | 758
|
| References | 759
|
| 22. Orthogonal Polynomials | 771
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| Mathematical Properties. 22.1. Definition of Orthogonal Polynomials | 773
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| 22.2. Orthogonality Relations | 774
|
| 22.3. Explicit Expressions | 775
|
| | 776
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| 22.4. Special Values. 22.5. Interrelations | 777
|
| | 778
|
| | 779
|
| | 780
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| 22.6. Differential Equations | 781
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| 22.7. Recurrence Relations | 782
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| 22.8. Differential Relations. 22.9. Generating Functions | 783
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| 22.10. Integral Representations | 784
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| 22.11. Rodrigues' Formula. 22.12. Sum Formulas. 22.13. Integrals Involving Orthogonal Polynomials | 785
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| 22.14. Inequalities | 786
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| 22.15. Limit Relations. 22.16. Zeros | 787
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| 22.17. Orthogonal Polynomials of a Discrete Variable. Numerical Methods. 22.18. Use and Extension of the Tables | 788
|
| | 789
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| 22.19. Least Square Approximations | 790
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| 22.20. Economization of Series | 791
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| References | 792
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| 23. Bernoulli and Euler Polynomials, Riemann Zeta Function | 803
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| Mathematical Properties. 23.1. Bernoulli and Euler Polynomials and the Euler-Maclaurin Formula | 804
|
| | 805
|
| | 806
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| 23.2. Riemann Zeta Function and Other Sums of Reciprocal Powers | 807
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| References | 808
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| 24. Combinatorial Analysis | 821
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| Mathematical Properties. 24.1. Basic Numbers. 24.1.1. Binomial Coefficients | 822
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| 24.1.2. Multinomial Coefficients | 823
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| 24.1.3. Stirling Numbers of the First Kind. 24.1.4. Stirling Numbers of the Second Kind | 824
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| 24.2. Partitions. 24.2.1. Unrestricted Partitions. 24.2.2. Partitions Into Distinct Parts | 825
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| 24.3. Number Theoretic Functions. 24.3.1. The Mobius Function. 24.3.2. The Euler Function | 826
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| 24.3.3. Divisor Functions. 24.3.4. Primitive Roots. References | 827
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| 25. Numerical Interpolation, Differentiation, and Integration | 875
|
| 25.1. Differences | 877
|
| 25.2. Interpolation | 878
|
| | 879
|
| | 880
|
| | 881
|
| 25.3. Differentiation | 882
|
| | 883
|
| | 884
|
| 25.4. Integration | 885
|
| | 886
|
| | 887
|
| | 888
|
| | 889
|
| | 890
|
| | 891
|
| | 892
|
| | 893
|
| | 894
|
| | 895
|
| 25.5. Ordinary Differential Equations | 896
|
| | 897
|
| References | 898
|
| | 899
|
| 26. Probability Functions | 925
|
| Mathematical Properties. 26.1. Probability Functions: Definitions and Properties | 927
|
| | 928
|
| | 929
|
| | 930
|
| 26.2. Normal or Gaussian Probability Function | 931
|
| | 932
|
| | 933
|
| | 934
|
| | 935
|
| 26.3. Bivariate Normal Probability Function | 936
|
| | 937
|
| 26.4. Chi-Square Probability Function | 940
|
| | 941
|
| | 942
|
| | 943
|
| 26.5. Incomplete Beta Function | 944
|
| | 945
|
| 26.6. F-(Variance-Ratio) Distribution Function | 946
|
| | 947
|
| 26.7. Student's t-Distribution | 948
|
| Numerical Methods. 26.8. Methods of Generating Random Numbers and Their Applications | 949
|
| | 950
|
| | 951
|
| | 952
|
| 26.9. Use and Extension of the Tables | 953
|
| | 954
|
| | 955
|
| References | 961
|
| | 962
|
| | 963
|
| | 964
|
| 27. Miscellaneous Functions | 997
|
| 27.1. Debye functions | 998
|
| 27.2. Planck's Radiation Function. 27.3. Einstein Functions | 999
|
| 27.4. Sievert Integral | 1000
|
| 27.5. $f_m(x)=\int_0^\infinity t^m e^{-t^2-x/t} dt$ and Related Integrals | 1001
|
| | 1002
|
| 27.6. $f(x)=\int_0^\infinity e^{-t^2}/(t+x) dt$ | 1003
|
| 27.7 Dilogarithm (Spence's Integral) | 1004
|
| 27.8. Clausen's Integral and Related Summations | 1005
|
| 27.9. Vector-Addition Coefficients | 1006
|
| | 1007
|
| | 1008
|
| | 1009
|
| | 1010
|
| 29. Laplace Transforms | 1019
|
| 29.1. Definition of the Laplace Transform. 29.2. Operations for the Laplace Transform | 1020
|
| 29.3. Table of Laplace Transforms | 1021
|
| | 1022
|
| | 1023
|
| | 1024
|
| | 1025
|
| | 1026
|
| | 1027
|
| | 1028
|
| 29.4. Table of Laplace-Stieltjes Transforms | 1029
|
| References | 1030
|
| Subject index A-B- | 1031
|
| Subject index -B-C- | 1032
|
| Subject index -C-D- | 1033
|
| Subject index -D-E- | 1034
|
| Subject index -E-F-G-H- | 1035
|
| Subject index -H-I- | 1036
|
| Subject index -I-J-K-L- | 1037
|
| Subject index -L-M- | 1038
|
| Subject index -M-N-O- | 1039
|
| Subject index -O-P- | 1040
|
| Subject index -P-Q-R-S- | 1041
|
| Subject index -S-T-U-V-W- | 1042
|
| Subject index -W-Z | 1043
|
| Index of Notations | 1044
|
| | 1045
|
| Notation -- Greek Letters. Miscellaneous Notations | 1046
|