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1. �� : Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Edited by M.Abramowitz and I.A.Stegun.
National Bureau of Standards, 1972. 1060 p. �� 17�.
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Title 1
Front Matter 2
Errata 2
Preface 3
Preface to the Ninth Printing 4
Foreword 5
Contents 7
Introduction 9
1. Introduction 9
2. Accuracy of the Tables 9
3. Auxiliary Functions and Arguments 10
4. Interpolation 10
5. Inverse Interpolation 12
6. Bivariate Interpolation 13
7. Generation of Functions from Recurrence Relations 13
8. Acknowledgements 14
1. Mathematical Constants � David S. Liepman 15
Table 1.1 � Mathematical Constants 16
2. Physical Constants and Conversion Factors � A. G. McNish 19
Table 2.1 � Common Units and Conversion Factors 20
Table 2.2 � Names and Conversion Factors for Electric and Magnetic Units 20
Table 2.3 � Adjusted Values of Constants 21
Table 2.4 � Miscellaneous Conversion Factors 22
Table 2.5 � Conversion Factors for Customary U.S. Units to Metric Units 22
Table 2.6 � Geodetic Constants 22
3. Elementary Analytical Methods � Milton Abramowitz 23
Elementary Analytical Methods 24
3.1. Binomial Theorem and Binomial Coefficients; Arithmetic and Geometric Progressions; Arithmetic, Geometric, Harmonic and Generalized Means 24
3.2. Inequalities 24
3.3. Rules for Differentiation and Integration 25
3.4. Limits, Maxima and Minima 27
3.5. Absolute and Relative Errors 28
3.6. Infinite Series 28
3.7. Complex Numbers and Functions 30
3.8. Algebraic Equations 31
3.9. Successive Approximation Methods 32
3.10. Theorems on Continued Fractions 33
Numerical Methods 33
3.11. Use and Extension of the Tables 33
References 37
Table 3.1 � Powers and Roots 38
4. Elementary Transcendental Functions � Ruth Zucker 79
Mathematical Properties 81
4.1. Logarithmic Function 81
4.2. Exponential Function 83
4.3. Circular Functions 85
4.4. Inverse Circular Functions 93
4.5. Hyperbolic Functions 97
4.6. Inverse Hyperbolic Functions 100
Numerical Methods 103
4.7. Use and Extension of the Tables 103
References 107
Table 4.1 � Common Logarithms (100<=x<=1350) 109
Table 4.2 � Natural Logarithms (0<=x<=2.1) 114
Table 4.3 � Radix Table of Natural Logarithms 128
Table 4.4 � Exponential Function (0<=|x|<=100 130
Table 4.5 � Radix Table of the Exponential Function 154
Table 4.6 � Circular Sines and Cosines for Radian Arguments (0<=x<=1000) 156
Table 4.7 � Radix Table of Circular Sines and Cosines 188
Table 4.8 � Circular Sines and Cosines for Large Radian Arguments (0<=x<=1000) 189
Table 4.9 � Circular Tangents, Cotangents, Secants and Cosecants for Radian Arguments (0<=x<=1.6) 200
Table 4.10 � Circular Sines and Cosines to Tenths of a Degree (0 degrees<=theta<=90 degrees) 203
Table 4.11 � Circular Tangents, Cotangents, Secants and Cosecants to Five Tenths of a Degree (0 degrees<=theta<=90 degrees) 212
Table 4.12 � Circular Functions for the Argument (pi/2) x (0<=x<=1) 214
Table 4.13 � Harmonic Analysis 216
Table 4.14 � Inverse Circular Sines and Tangents (0<=x<=1) 217
Table 4.15 � Hyperbolic Functions (0<=x<=10) 227
Table 4.16 � Exponential and Hyperbolic Functions for the Argument pi x (0<=x<=1) 233
Table 4.17 � Inverse Hyperbolic Functions (0<=x<=infinity) 235
Table 4.18 � Roots x_n of cos x_n cosh x_n = 1 or -1 237
Table 4.19 � Roots x_n of tan x_n = lambda x_n 238
Table 4.20 � Roots x_n of cot x_n = lambda x_n 239
5. Exponential Integral and Related Functions � Walter Gautschi, William F. Cahill 241
Mathematical Properties 242
5.1. Exponential Integral 242
5.2. Sine and Cosine Integrals 245
Numerical Methods 247
5.3. Use and Extension of the Tables 247
References 249
Table 5.1 � Sine, Cosine and Exponential Integrals 252
Table 5.2 � Sine, Cosine and Exponential Integrals for Large Arguments 257
Table 5.3 � Sine and Cosine Integrals for Arguments pi x 258
Table 5.4 � Exponential Integral E_n(x) 259
Table 5.5 � Exponential Integral E_n(x) for Large Arguments 262
Table 5.6 � Exponential Integral for Complex Arguments 263
Table 5.7 � Exponential Integral for Small Complex Arguments 265
6. Gamma Function and Related Functions � Philip J. Davis 267
Mathematical Properties 269
6.1. Gamma Function 269
6.2. Beta Function 272
6.3. Psi (Digamma) Function 272
6.4. Polygamma Functions 274
6.5. Incomplete Gamma Function 274
6.6. Incomplete Beta Function 277
Numerical Methods 277
6.7. Use and Extension of the Tables 277
6.8. Summation of Rational Series by Means of Polygamma Functions 278
References 279
Table 6.1 � Gamma, DiGamma and TriGamma Functions 281
Table 6.2 � TetraGamma and PentaGamma Functions 285
Table 6.3 � Gamma and DiGamma Functions for Integer and Half-Integer Values 286
Table 6.4 � Logarithms of the Gamma Function 288
Table 6.5 � Auxiliary Functions for Gamma and DiGamma Functions 290
Table 6.6 � Factorials for Large Arguments 290
Table 6.7 � Gamma Function for Complex Arguments 291
Table 6.8 � DiGamma Function for Complex Arguments 302
7. Error Function and Fresnel Integrals � Walter Gautschi 309
Mathematical Properties 311
7.1. Error Function 311
7.2. Repeated Integrals of the Error Function 313
7.3. Fresnel Integrals 314
7.4. Definite and Indefinite Integrals 316
Numerical Methods 318
7.5. Use and Extension of the Tables 318
References 322
Table 7.1 � Error Function and its Derivative 324
Table 7.2 � Derivative of the Error Function 326
Table 7.3 � Complementary Error Function 330
Table 7.4 � Repeated Integrals of the Error Function 331
Table 7.5 � Dawson's Integral 333
Table 7.6 � 3/Gamma(1/3) Integral from 0 to x of e**(-t**2) dt 334
Table 7.7 � Fresnel Integrals 335
Table 7.8 � Auxiliary Functions 337
Table 7.9 �Error Function for Complex Arguments 339
Table 7.10 � Complex Zeros of the Error Function 343
Table 7.11 � Complex Zeros of Fresnel Integrals 343
Table 7.12 � Maxima and Minima of Fresnel Integrals 343
8. Legendre Functions � Irene A. Stegun 345
Mathematical Properties 346
8.1. Differential Equation 346
8.2. Relations Between Legendre Functions 347
8.3. Values on the Cut 347
8.4. Explicit Expressions 347
8.5. Recurrence Relations 347
8.6. Special Values 348
8.7. Trigonometric Expansions (0 < theta < pi) 349
8.8. Integral Representations 349
8.9. Summation Formulas 349
8.10. Asymptotic Expansions 349
8.11. Toroidal Functions (or Ring Functions) 350
8.12. Conical Functions 351
8.13. Relation to Elliptic Integrals 351
8.12. Integrals 351
Numerical Methods 353
8.15. Use and Extension of the Tables 353
References 354
Table 8.1 � Legendre Function � First Kind P_n(x) 356
Table 8.2 � Derivative of the Legendre Function � First Kind P'_n(x) 358
Table 8.3 � Legendre Function � Second Kind Q_n(x) 360
Table 8.4 � Derivative of the Legendre Function � Second Kind Q'_n(x) 362
Table 8.5 � Legendre Function � First Kind P_n(x) 364
Table 8.6 � Derivative of the Legendre Function � First Kind P_n(x) 365
Table 8.7 � Legendre Function � Second Kind Q_n(x) 366
Table 8.8 � Derivative of the Legendre Function � Second Kind Q'_n(x) 367
9. Bessel Functions of Integer Order � F. W. J. Olver 369
Mathematical Properties 372
Notation 372
Bessel Functions J and Y 372
9.1. Definitions and Elementary Properties 372
9.2. Asymptotic Expansions for Large Arguments 378
9.3. Asymptotic Expansions for Large Orders 379
9.4. Polynomial Approximations 383
9.5. Zeros 384
Modified Bessel Functions I and K 388
9.6. Definitions and Properties 388
9.7. Asymptotic Expansions 391
9.8. Polynomial Approximations 392
Kelvin Functions 393
9.9. Definitions and Properties 393
9.10. Asymptotic Expansions 395
9.11. Polynomial Approximations 398
Numerical Methods 399
9.12. Use and Extension of the Tables 399
References 402
Table 9.1 � Bessel Functions � Orders 0, 1 and 2 404
Table 9.2 � Bessel Functions � Orders 3�9 412
Table 9.3 � Bessel Functions � Orders 10, 11, 20 and 21 416
Table 9.4 � Bessel Functions � Various Orders 421
Table 9.5 � Zeros and Associated Values of Bessel Functions and Their Derivatives 423
Table 9.6 � Bessel Functions J_0(j_{0,s} x) 427
Table 9.7 � Bessel Functions � Miscellaneous Zeros 428
Table 9.8 � Modified Bessel Functions of Orders 0, 1 and 2 430
Table 9.9 � Modified Bessel Functions � Orders 3�9 437
Table 9.10 � Modified Bessel Functions � Orders 10, 11, 20 and 21 439
Table 9.11 � Modified Bessel Functions � Various Orders 442
Table 9.12 � Kelvin Functions � Orders 0 and 1 444
10. Bessel Functions of Fractional Order � H. A. Antosiewicz 450
Mathematical Properties 452
10.1. Spherical Bessel Functions 452
10.2. Modified Spherical Bessel Functions 457
10.3. Riccati-Bessel Functions 459
10.4. Airy Functions 460
Numerical Methods 466
10.5. Use and Extension of the Tables 466
References 469
Table 10.1 � Spherical Bessel Functions � Orders 0, 1 and 2 471
Table 10.2 � Spherical Bessel Functions � Orders 3�10 473
Table 10.3 � Spherical Bessel Functions � Orders 20 and 21 477
Table 10.4 � Spherical Bessel Functions � Modulus and Phase � Orders 9, 10, 20 and 21 478
Table 10.5 � Spherical Bessel Functions � Various Orders 479
Table 10.6 � Zeros of Bessel Functions of Half-Integer Order 481
Table 10.7 � Zeros of the Derivative of Bessel Functions of Half-Integer Order 482
Table 10.8 � Modified Spherical Bessel Functions � Orders 0, 1 and 2 483
Table 10.9 � Modified Spherical Bessel Functions � Orders 9 and 10 484
Table 10.10 � Modified Spherical Bessel Functions � Various Orders 487
Table 10.11 � Airy Functions 489
Table 10.12 � Integrals of Airy Functions 492
Table 10.13 � Zeros and Associated Values of Airy Functions and Their Derivatives 492
11. Integrals of Bessel Functions � Yudell L. Luke 493
Mathematical Properties 494
11.1. Simple Integrals of Bessel Functions 494
11.2. Repeated Integrals of J_n(z) and K_0(z) 496
11.3. Reduction Formulas for Indefinite Integrals 497
11.4. Definite Integrals 499
Numerical Methods 502
11.5. Use and Extension of the Tables 502
References 504
Table 11.1 � Integrals of Bessel Functions 506
Table 11.2 � Integrals of Bessel Functions 508
12. Struve Functions and Related Functions � Milton Abramowitz 509
Mathematical Properties 510
12.1. Struve Function H_nu(z) 510
12.2. Modified Struve Function L_nu(z) 512
12.3. Anger and Weber Functions 512
Numerical Methods 513
12.4. Use and Extension of the Tables 513
References 514
Table 12.1 � Struve Functions 515
Table 12.2 � Struve Functions for Large Arguments 516
13. Confluent Hypergeometric Functions � Lucy Joan Slater 517
Mathematical Properties 518
13.1. Definitions of Kummer and Whittaker Functions 518
13.2. Integral Representations 519
13.3. Connections with Bessel Functions 520
13.4. Recurrence Relations and Differential Properties 520
13.5. Asymptotic Expansions and Limiting Forms 522
13.6. Special Cases 523
13.7. Zeros and Turning Values 524
Numerical Methods 525
13.8. Use and Extension of the Tables 525
13.9. Calculation of Zeros and Turning Points 527
13.10. Graphing M(a, b, x) 527
References 528
Table 13.1 � Confluent Hypergeometric Function M(a, b, x) 530
Table 13.2 � Zeros of M(a, b, x) 549
14. Coulomb Wave Functions � Milton Abramowitz 551
Mathematical Properties 552
14.1. Differential Equation, Series Expansion 552
14.2. Recurrence and Wronskian Relations 553
14.3. Integral Representations 553
14.4. Bessel Function Expansions 553
14.5. Asymptotic Expansions 554
14.6. Special Values and Asymptotic Behaviour 556
Numerical Methods 557
14.7. Use and Extension of the Tables 557
References 558
Table 14.1 � Coulomb Wave Functions of Order Zero 560
Table 14.2 � C_0(eta) = e**{-1/2 pi eta} | Gamma (1 + i eta) | 568
15. Hypergeometric Functions � Fritz Oberhettinger 569
Mathematical Properties 570
15.1. Gauss Series, Special Elementary Cases, Special Values of the Argument 570
15.2. Differentiation Formulas and Gauss' Relations for Contiguous Functions 571
15.3. Integral Representations and Transformation Formulas 572
15.4. Special Cases of F(a, b; c; z), Polynomials and Legendre Functions 575
15.5. The Hypergeometric Differential Equation 576
15.6. Riemann's Differential Equation 578
15.7. Asymptotic Expansions 579
References 579
16. Jacobian Elliptic Functions and Theta Functions � L. M. Milne-Thomson 581
Mathematical Properties 583
16.1. Introduction 583
16.2. Classification of the Twelve Jacobian Elliptic Functions 584
16.3. Relation of the Jacobian Functions to the Copolar Trio 584
16.4. Calculation of the Jacobian Functions by Use of the Arithmetic-Geometric Mean (A.G.M) 585
16.5. Special Arguments 585
16.6. Jacobian Functions when m=0 or 1 585
16.7. Principal Terms 586
16.8. Change of Argument 586
16.9. Relations Between the Squares of the Functions 587
16.10. Change of Parameter 587
16.11. Reciprocal Parameter (Jacobi's Real Transformation) 587
16.12. Descending Landen Transformation (Gauss' Transformation) 587
16.13. Approximation in Terms of Circular Functions 587
16.14. Ascending Landen Transformation 587
16.15. Approximation in Terms of Hyperbolic Functions 588
16.16. Derivatives 588
16.17. Addition Theorems 588
16.18. Double Arguments 588
16.19. Half Arguments 588
16.20. Jacobi's Imaginary Transformation 588
16.21. Complex Arguments 589
16.22. Leading Terms of the Series in Ascending Powers of u 589
16.23. Series Expansion in Terms of the Nome q 589
16.24. Integrals of the Twelve Jacobian Elliptic Functions 589
16.25. Notation for the Integrals of the Squares of the Twelve Jacobian Elliptic Functions 590
16.26. Integrals in Terms of the Elliptic Integral of the Second Kind 590
16.27. Theta Functions; Expansions in Terms of the Nome q 590
16.28. Relations Between the Squares of the Theta Functions 590
16.29. Logarithmic Derivatives of the Theta Functions 590
16.30. Logarithms of Theta Functions of Sum and Difference 591
16.31. Jacobi's Notation for Theta Functions 591
16.32. Calculation of Jacobi's Theta Function Theta(u|m) by Use of the Arithmetic-Geometric Mean 591
16.33. Addition of Quarter-Periods to Jacobi's Eta and Theta Functions 591
16.34. Relation of Jacobi's Zeta Function to the Theta Functions 592
16.35. Calculation of Jacobi's Zeta Function Z(u|m) by Use of the Arithmetic-Geometric Mean 592
16.36. Neville's Notation for Theta Functions 592
16.37. Expression as Infinite Products 593
16.38. Expression as Infinite Series 593
Numerical Methods 593
16.39. Use and Extension of the Tables 593
References 595
Table 16.1 � Theta Functions 596
Table 16.2 � Logarithmic Derivatives of Theta Functions 598
17. Elliptic Integrals � L. M. Milne-Thomson 601
Mathematical Properties 603
17.1. Definition of Elliptic Integrals 603
17.2. Canonical Forms 603
17.3. Complete Elliptic Integrals of the First and Second Kinds 604
17.4. Incomplete Elliptic Integrals of the First and Second Kinds 606
17.5. Landen's Transformation 611
17.6. The Process of the Arithmetic-Geometric Mean 612
17.7. Elliptic Integrals of the Third Kind 613
Numerical Methods 614
17.8. Use and Extension of the Tables 614
References 620
Table 17.1 � Complete Elliptic Integrals of the First and Second Kinds and the Nome q With Argument the Parameter m 622
Table 17.2 � Complete Elliptic Integrals of the First and Second Kinds and the Nome q With Argument the Modular Angle alpha 624
Table 17.3 � Parameter m With Argument K'(m)/K(m) 626
Table 17.4 � Auxiliary Functions for Computation of the Nome q and the Parameter m 626
Table 17.5 � Elliptical Integral of the First Kind F(phi\alpha) 627
Table 17.6 � Elliptical Integral of the Second Kind E(phi\alpha) 630
Table 17.7 � Jacobian Zeta Function Z(phi\alpha) 633
Table 17.8 � Heuman's Lambda Function Lambda_0(phi\alpha) 636
Table 17.9 � Elliptical Integral of the Third Kind Pi(n; phi\alpha) 639
18. Weierstrass Elliptic and Related Functions � Thomas H. Southard 641
Mathematical Properties 643
18.1. Definitions, Symnbolism, Restrictions and Convertions 643
18.2. Homogeneity Relations, Reductions Formulas and Processes 645
18.3. Special Values and Relations 647
18.4. Addition and Multiplication Formulas 649
18.5. Series Expansions 649
18.6. Derivatives and Differential Equations 654
18.7. Integrals 655
18.8. Conformal Mapping 656
18.9. Relations with Complete Elliptic Integrals K and K' and Their Parameter m and with Jacobi's Elliptic Functions 663
18.10. Relations with Theta Functions 664
18.11. Expressing any Elliptic Function in Terms of P and P' 665
18.12. Case Delta = 0 665
18.13. Equianharmonic Case (g_2=0, g_3=1) 666
18.14. Lemniscatic Case (g_2=1, g_3=0) 672
18.15. Pseudo-Lemniscatic Case (g_2=-1, g_3=0) 676
Numerical Methods 677
18.16. Use and Extension of the Tables 677
References 684
Table 18.1 � Table for Obtaining Periods for Invariants g_2 and g_3 687
Table 18.2 � Table for Obtaining P, P' and zeta on 0x and 0y 688
Table 18.3 � Invariants and Values at Half-Periods (1<=a<=infinity) 694
19. Parabolic Cylinder Functions � J. C. P. Miller 699
Mathematical Properties 700
19.1. The Parabolic Cylinder Functions 700
19.2. Power Series in x 700
19.3. Standard Solutions 701
19.4. Wronskian and Other Relations 701
19.5. Integral Representations 701
19.6. Recurrence Relations 702
19.7. Expressions in Terms of Airy Functions 703
19.8. Expansions for x Large and a Moderate 703
19.9. Expansions for a Large and x Moderate 703
19.10. Darwins Expansions 703
19.11. Modulus and Phase 704
19.12. Connection with Confluent Hypergeometric Functions 705
19.13. Connection with Hermite Polynomials and Functions 705
19.14. Connection with Probability Integrals and Dawson's Integral 705
19.15. Explicit Formula in Terms of Bessel Functions when 2a is an Integer 706
19.16. Power Series in x 706
19.17. Standard Solutions 706
19.18. Wronskian and Other Relations 707
19.19. Integral Representations 707
19.20. Expressions in Terms of Airy Functions 707
19.21. Expansions for x Large and a Moderate 707
19.22. Expansions for a Large and x Moderate 708
19.23. Darwin's Expansions 708
19.24. Modulus and Phase 709
19.25. Connections with Other Functions 709
19.26. Zeros 710
19.27. Bessel Functions of Order +-1/4, +-3/4 as Parabolic Cylinder Functions 711
Numerical Methods 711
19.28. Use and Extension of the Tables 711
References 714
Table 19.1 � U(a, x) and V(a, x) (0<=x<=5) 716
Table 19.2 � W(a, +-x) (0<=x<=5) 726
Table 19.3 � Auxiliary Functions 734
20. Mathieu Functions � Gertrude Blanch 735
Mathematical Properties 736
20.1. Mathieu's Equation 736
20.2. Determination of Characteristic Values 736
20.3. Floquet's Theorem and its Consequences 741
20.4. Other Solutions of Mathieu's Equation 744
20.5. Properties of Orthogonality and Normalization 746
20.6. Solutions of Mathieu's Modified Equation for Integral nu 746
20.7. Representations by Integrals and Some Integral Equations 749
20.8. Other Properties 752
20.9. Asymptotic Representations 754
20.10. Comparative Notations 758
References 759
Table 20.1 � Characteristic Values, Joining Factors, Some Critical Values (0<=q<=infinity) 762
Table 20.2 � Coefficients A_m and B_m 764
21. Spheroidal Wave Functions � Arnold N. Lowan 765
Mathematical Properties 766
21.1. Definition of Elliptical Coordinates 766
21.2. Definition of Prolate Spheroidal Coordinates 766
21.3. Definition of Oblate Spheroidal Coordinates 766
21.4. Laplacian in Spheroidal Coordinates 766
21.5. Wave Equation in Prolate and Olate Spheroidal Coordinates 766
21.6. Differential Equations for Radial and Angular Spheroidal Wave Equations 767
21.7. Prolate Angular Functions 767
21.8. Oblate Angular Functions 770
21.9. Radial Spheroidal Wave Functions 770
21.10. Joining Factors for Prolate Spheroidal Wave Functions 771
21.11. Notation 772
References 773
Table 21.1 � Eigenvalues � Prolate and Oblate 774
Table 21.2 � Angular Functions � Prolate and Oblate 780
Table 21.3 � Prolate Radial Functions � First and Second Kinds 782
Table 21.4 � Oblate Radial Functions � First and Second Kinds 783
Table 21.5 � Prolate Joining Factors � First Kind 783
22. Orthogonal Polynomials � Urs W. Hochstrasser 785
Mathematical Properties 787
22.1. Definition of Orthogonal Polynomials 787
22.2. Orthogonality Relations 788
22.3. Explicit Expressions 789
22.4. Special Values 791
22.5. Interrelations 791
22.6. Differential Equations 795
22.7. Recurrence Relations 796
22.8. Differential Relations 797
22.9. Generating Functions 797
22.10. Integral Representations 798
22.11. Rodrigues' Formula 799
22.12. Sum Formulas 799
22.13. Integrals Involving Orthogonal Polynomials 799
22.14. Inequalities 800
22.15. Limit Relations 801
22.16. Zeros 801
22.17. Orthogonal Polynomials of a Discrete Variable 802
Numerical Methods 802
22.18. Use and Extension of the Tables 802
22.19. Least Square Approximations 804
22.20. Economization of Series 805
References 806
Table 22.1 � Coefficients for the Jacobi Polynomials P**(alpha, beta)_n(x) 807
Table 22.2 � Coefficients for the Ultraspherical Polynomials C**(alpha)_n(x) and for x**n in Terms of C**(alpha)_m(x) 808
Table 22.3 � Coefficients for the Chebyshev Polynomials T_n(x) and for x**n in Terms of T_m(x) 809
Table 22.4 � Values of the Chebyshev Polynomials T_n(x) 809
Table 22.5 � Coefficients for the Chebyshev Polynomials U_n(x) and for x**n in Terms of U_m(x) 810
Table 22.6 � Values of the Chebyshev Polynomials U_n(x) 810
Table 22.7 � Coefficients for the Chebyshev Polynomials C_n(x) and for x**n in Terms of C_m(x) 811
Table 22.8 � Coefficients for the Chebyshev Polynomials S_n(x) and for x**n in Terms of S_m(x) 811
Table 22.9 � Coefficients for the Legendre Polynomials P_n(x) and for x**n in Terms of P_m(x) 812
Table 22.10 � Coefficients for the Laguerre Polynomials L_n(x) and for x**n in Terms of L_m(x) 813
Table 22.11 � Values of the Laguerre Polynomials L_n(x) 814
Table 22.12 � Coefficients for the Hermite Polynomials H_n(x) and for x**n in Terms of H_m(x) 815
Table 22.13 � Values of the Hermite Polynomials H_n(x) 816
23. Bernoulli and Euler Polynomials � Riemann Zeta Function � Emilie V. Haynsworth, Karl Goldberg 817
Mathematical Properties 818
23.1. Bernoulli and Euler Polynomials and the Euler-Maclaurin Formula 818
23.2. Riemann Zeta Function and Other Sums of Reciprocal Powers 821
References 822
Table 23.1 � Coefficients of the Bernoulli and Euler Polynomials B_n(x) and E_n(x) 823
Table 23.2 � Bernoulli and Euler Numbers B_n and E_n 824
Table 23.3 � Sums of Reciprocal Powers 825
Table 23.4 � Sums of Positive Powers 827
Table 23.5 � x**n/n! 832
24. Combinatorial Analysis � K. Goldberg, M. Newman, E. Haynsworth 835
24.1. Basic Numbers 836
24.2. Partitions 839
24.3. Number Theoretic Functions 840
References 841
Table 24.1 � Binomial Coefficients 842
Table 24.2 � Multinomials (Including a List of Partitions) 845
Table 24.3 � Stirling Numbers of the First Kind 847
Table 24.4 � Stirling Numbers of the Second Kind 849
Table 24.5 � Number of Partitions and Partitions Into Distinct Parts 850
Table 24.6 � Arithmetic Functions 854
Table 24.7 � Factorizations 858
Table 24.8 � Primitive Roots, Factorization of p-1 878
Table 24.9 � Primes 884
25. Numerical Interpolation, Differentiation and Integration � Philip J. Davis, Ivan Polonsky 889
25.1. Differences 891
25.2. Interpolation 892
25.3. Differentiation 896
25.4. Integration 899
25.5. Ordinary Differential Equations 910
References 912
Table 25.1 � n-Point Langrangian Interpolation Coefficients (3<=n<=8) 914
Table 25.2 � n-Point Coefficients for k-th Order Differentiation (1<=k<=5) 928
Table 25.3 � n-Point Langrangian Integration Coefficients (3<=n<=10) 929
Table 25.4 � Abscissas and Weight Factors for Gaussian Integration (2<=n<=96) 930
Table 25.5 � Abscissas for Equal Weight Chebyshev Integration (2<=n<=9) 934
Table 25.6 � Abscissas and Weight Factors for Lobatto Integration (3<=n<=10) 934
Table 25.7 � Abscissas and Weight Factors for Gaussian Integration for Integrands with a Logarithmic Singularity (2<=n<=4) 934
Table 25.8 � Abscissas and Weight Factors for Gaussian Integration of Moments (1<=n<=8) 935
Table 25.9 � Abscissas and Weight Factors for Laguerre Integration (2<=n<=15) 937
Table 25.10 � Abscissas and Weight Factors for Hermite Integration (2<=n<=20) 938
Table 25.11 � Coefficients for Filon's Quadrature Formula (0<=theta<=1) 938
26. Probability Functions � Marvin Zelen, Norman C. Severo 939
Mathematical Properties 941
26.1. Probability Functions: Definitions and Properties 941
26.2. Normal or Gaussian Probability Function 945
26.3. Bivariate Normal Probability Function 950
26.4. Chi-Square Probability Function 954
26.5. Incomplete Beta Function 958
26.6. F-(Variance-Ratio) Distribution Function 960
26.7. Student's t-Distribution 962
Numerical Methods 963
26.8. Methods of Generating Random Numbers and Their Applications 963
26.9. Use and Extension of the Tables 967
References 975
Table 26.1 � Normal Probability Function and Derivatives (0<=x<=5) 980
Table 26.2 � Normal Probability Function for Large Arguments (5<=x<=500) 987
Table 26.3 � Higher Derivatives of the Normal Probability Function (0<=x<=5) 988
Table 26.4 � Normal Probability Function � Values of Z(x) in Terms of P(x) and Q(x) 989
Table 26.5 � Normal Probability Function � Values of x in Terms of P(x) and Q(x) 990
Table 26.6 � Normal Probability Function � Values of x for Extreme Values of P(x) and Q(x) 991
Table 26.7 � Probability Integral of chi**2-Distribution, Incomplete Gamma Function, Cumulative Sums of the Poisson Distribution 992
Table 26.8 � Percentage Points of the chi**2-Distribution � Values of chi**2 in Terms of Q and nu 998
Table 26.9 � Percentage Points of the F-Distribution � Values of chi**2 in Terms of Q, nu_1, nu_2 1000
Table 26.10 � Percentage Points of the t-Distribution � Values of t in Terms of A and nu 1004
Table 26.11 � 2500 Five Digit Random Numbers 1005
27. Miscellaneous Functions � Irene A. Stegun 1011
27.1. Debye Functions 1012
27.2. Planck's Radiation Function 1013
27.3. Einstein Functions 1013
27.4. Sievert Integral 1014
27.5. Unnamed and Related Integrals 1015
27.6. Unnamed Integral 1017
27.7. Dilogarithm (Spence's Integral) 1018
27.8. Clausen's Integral and Related Summations 1019
27.9. Vector-Addition Coefficients 1020
28. Scales of Notation � S. Peavy, A. Schopf 1025
Representation of Numbers 1026
Numerical Methods 1027
References 1029
Table 28.1 � 2**+-n in Decimal 1030
Table 28.2 � 2**x in Decimal 1031
Table 28.3 � 10**+-n in Octal 1031
Table 28.4 � n log_10 2, n log_2 10 in Decimal 1031
Table 28.5 � Addition and Multiplication Tables 1031
Table 28.6 � Mathematical Constants in Octal Scale 1031
29. Laplace Transforms 1033
29.1. Definition of the Laplace Transform 1034
29.2. Operations for the Laplace Transform 1034
29.3. Table of Laplace Transforms 1035
29.4. Table of Laplace-Stieltjes Transform 1043
References 1044
Subject Index 1045
Index of Notations 1058
Notation � Greek Letters 1060
Miscellaneous Notations 1060
��:
�� 5
�� 7
�� 1. �� 12
�� 2. �� 15
�� 2.1. �� 15
�� 2.2. �� 16
�� 2.3. �� 16
�� 2.4. �� 17
�� 2.5. �� , �� , � �� 18
�� 2.6. �� 18
�� 3. �� 19
3.1. �� ; �� ; ��, ��, �� 19
3.2. �� 20
3.3. �� 21
3.4. ��, �� 23
3.3. �� 23
3.6. �� 24
3.7. �� 26
3.8. �� 27
3.9. �� 27
3.10. �� 28
�� 29
�� 31
�� 4. �� . ��, ��, �� 33
4.1. �� 33
4.2. �� 35
4.3. �� 37
4.4. �� 44
4.5. �� 48
4.6. �� 50
�� 53
�� 5. �� 55
5.1. �� 56
5.2. �� 59
�� 61
�� 5.1. �� , �� 63
�� 5.2. �� , �� 68
�� 5.3. �� πx 69
�� 5.4. �� 70
�� 5.5. �� 73
�� 5.6. �� 74
�� 5.7. �� 76
�� 77
�� 6. ��-�� 80
6.1. ��-�� 81
6.2. ��-�� 84
6.3. ��-�� (��-��) 84
6.4. ��-�� 85
6.5. �� -�� 86
6.6. �� -�� 89
�� 89
6.7. �� 89
6.8. �� -�� 90
�� 6.1. ��-, ��- � ��-�� 91
�� 6.2. ��- � ��-�� 95
�� 6.3. ��- � ��-�� 96
�� 6.4. �� -�� 98
�� 6.5. �� - � ��-�� 100
�� 6.6. �� 100
�� 6.7. ��-�� 101
�� 6.8. ��-�� 112
�� 118
�� 7. �� 119
7.1. �� 120
7.2. �� 122
7.3. �� 123
7.4. �� 125
�� 127
�� 7.1. �� 131
�� 7.2. �� 133
�� 7.3. �� 137
�� 7.4. �� 138
�� 7.5. �� 140
�� 7.6. (3/�(1/3)) 141
�� 7.7. �� 142
�� 7.8. �� 144
�� 7.9. �� 146
�� 7.10. �� 150
�� 7.11. �� 150
�� 7.12. �� 150
�� 151
�� 8. �� 153
�� 154
8.1. �� 154
8.2. �� 155
8.3. �� 155
8.4. �� 156
8.5. �� 156
8.6. �� 156
8.7. �� 157
8.8. �� 157
8.9. �� 158
8.10. �� 158
8.11. �� (�� ) 159
8.12. �� 159
8.13. �� 159
8.14. �� 160
�� 162
�� 8.1. �� Pn(�) (x≤1) 163
�� 8.2. �� Вn(�) (x≤1) 165
�� 8.3. �� Qn(x) (x≤1) 167
�� 8.4. �� Q�n(x) (x≤1) 169
�� 8.5. �� Pn(�) (x≥1) 171
�� 8.6. �� n(�) (x≥1) 172
�� 8.7. �� Qn(x)(x≥1) 173
�� 8.8. �� Q'n(x) (x≥1) 174
�� 175
�� 9. �� 177
�� 179
�� J � Y 180
9.1. �� 180
9.2. �� 185
9.3. �� 187
9.4. �� 191
9.5. �� 191
�� I � K 195
9.6. �� 195
9.7. �� 199
9.8. �� 199
�� 200
9.9. �� 200
9.10. �� 202
9.11. �� 205
�� 206
�� 9.1. �� 0, 1 � 2 208
�� 0, 1 � 2 214
�� (0<�<2) 215
�� 9.2. �� 3�9 216
�� 9.3. �� 10,11, 20 � 21 (0<�<20) 220
�� 10,11, 20 � 21 224
�� 9.4.�� (0<n<100) 225
�� 9.5. �� 227
�� 9.6. �� J0(j0,s x) 231
�� 9.7. �� , �� 232
�� 9.8. �� 0, 1 � 2 234
�� 240
�� 240
�� 9.9. �� 3�9 241
�� 9.10. �� 10,11,20 � 21 243
�� 245
�� 9.11. �� 246
�� 9.12. �� 0 � 1 248
�� 248
�� 250
�� 250
�� 254
�� 10. �� 254
10.1. �� 256
10.2. �� 261
10.3. �� 263
10.4. �� 264
�� 270
�� 10.1. �� 0,1 � 2 (0<x<10) 273
�� 10.2. �� 3 �10 (0<x<10) 275
�� 10.3. �� 20 � 21 (0<x<25) 279
�� 10.4. �� 9,10,20 � 21 280
�� 10.5. �� (0<n<100) 281
�� 10.6. �� (0<n<19) 283
�� 10.7. �� (0<n<19) 284
�� 10.8. �� 0, 1 � 2 (0<x<<5) 285
�� 10.9. �� 9 � 10 (0<x<∞) 286
�� 10.10. �� (0<n<100) 289
�� 10.11. �� (0<x<∞) 291
�� 10.12. �� (0<x<10) 294
�� 10.13. �� (1<s<10) 294
�� 295
�� 11. �� 297
11.1. �� 297
11.2. �� Jn(z) � K0(z) 300
11.3. �� 300
11.4. �� 302
�� 305
�� 11.1. �� 308
�� 11.2. �� 310
�� 311
�� 12. �� 313
12.1. �� Hv(z) 313
12.2. �� Lv(z) 315
12.3. �� 316
�� 317
�� 12.1. �� (0≤�<∞) 318
�� 12.2. �� 319
�� 320
�� 13. �� 321
13.1. �� 321
13.2. �� 323
13.3. �� 323
13.4. �� 324
13.5. �� 325
13.6. �� 326
13.7. �� 328
�� 329
13.8. �� 329
13.9. �� 330
13.10. �� (a,b,x) 331
�� 13.1. �� (a,b,x) 333
�� 13.2. �� (a,b,x) 352
�� 353
�� 14. �� 354
14.1. �� , �� 354
14.2. �� 355
14.3. �� 355
14.4. �� 356
14.5. �� 356
14.6. �� 358
�� 359
�� 14.1. �� 360
�� 14.2. C0(η) 368
�� 369
�� 15. �� 370
15.1. �� , �� , �� 370
15.2. �� 372
15.3. �� 373
15.4. �� F(a,b;c;z) 375
15.5. �� 377
15.6. �� 378
15.7. �� 379
�� 379
�� 16. �� -�� 380
16.1. �� 381
16.2. �� 382
16.3. �� sn u, cn u, dn u 382
16.4. �� -�� (�.�.�.) 383
16.5. �� 383
16.6. �� m=0 � m=1 383
16.7. �� 384
16.8. �� 384
16.9. �� 384
16.10. �� 385
16.11. �� (�� ) 385
16.12. �� (�� ) 385
16.13. �� 385
16.14. �� 385
16.15. �� 386
16.16. �� 386
16.17. �� 386
16.18. �� 387
16.19. �� 387
16.20. �� 387
16.21. �� 387
16.22. �� u 387
16.23. �� q � �� v 388
16.24. �� 388
16.25. �� 389
16.26. �� 389
16.27. ��-��; �� q 389
16.28. �� -�� 390
16.29. �� -�� 390
16.30. �� -�� 390
16.31. �� -�� 390
16.32. �� -�� θ (u|m) � �� -�� (�.�.�.) 390
16.33. �� - � ��-�� 391
16.34. �� -�� -�� 391
16.35. �� -�� Ζ(u|m) �� -�� (�.�.�.) 391
16.36. �� -�� 392
16.37. �� 392
16.38. �� 392
�� 393
�� 16.1. ��-�� 396
�� 16.2. �� -�� 398
�� 400
�� 17. �� 401
17.1 �� 402
17.2. �� 402
17.3. �� 403
17.4. �� 405
17.5. �� 412
17.6. �� -�� 413
17.7. �� 413
�� 415
�� 17.1. �� q �� m 422
�� 17.2. �� q �� α 424
�� 17.3. �� m �� ʒ(m)/K(m) 426
�� 17.4. �� q � �� m 426
�� 17.5. �� F(φ\α) 427
�� 17.6. �� E(φ\α) 430
�� 17.7. ��-�� Ζ(φ\α) 433
�a��a 17.8. ��-�� Λ0(φ\α) 436
�� 17.9. �� (n; φ\α) 439
�� 441
�� 18. �� 442
18.1. ��, ��, �� 442
18.2. �� 444
18.3. �� 445
18.4. �� 447
18.5. �� 447
18.6. �� 452
18.7. �� 453
18.8. �� 453
18.9. �� ', � �� m � � �� 460
18.10. �� -�� 461
18.11. �� Ԓ 462
18.12. �� Δ=0 (�>0) 462
18.13. �� (g2=0, g3=1) 463
18.14. �� (g2=1, g3=0) 468
18.15. �� (g2=-1, g3=0) 473
�� 474
�� 18.1. �� g2 � g3 482
�� 18.2. �� , Ԓ � ζ �� ΟΥ (�� ; �� ) 483
�� 18.3. �� (1<�< ∞) (�� ) 489
�� 493
�� 19. �� 494
19.1. �� . �� 494
19.2�19.6. �� , �� , �� , �� , �� 495
19.7�19.11. �� 498
19.12�19.15. �� 501
19.16�19.19. �� , �� , �� , �� 503
19.20�19.24. �� 504
19.25. �� 507
19.26. �� 507
19.27. �� 1/4, �3/4 �� 509
�� 509
�� 19.1. U(a,�) � V(a,�) 512
�� 19.2. W(a, ��) 522
�� 19.3. �� 530
�� 531
�� 20. �� 532
20.1. �� 532
20.2. �� 533
20.3. �� 537
20.4. �� 540
20.5. �� 542
20.6. �� ν 542
20.7. �� 545
20.8. �� 548
20.9. �� 549
20.10. �� 552
�� 20.1. �� , �� , �� (0<q<∞) 554
�� 20.2. �� Am � Bm 556
�� 557
�� 21. �� 559
21.1. �� 559
21.2. �� 560
21.3. �� 560
21.4. �� 560
21.5. �� 560
21.6. �� 561
21.7. �� 561
21.8. �� 564
21.9. �� 564
21.10. �� 564
21.11. �� 565
�� 21.1. �� 567
�� 21.2. �� 573
�� 21.3. �� 575
�� 21.4. �� 576
�� 21.5. �� 576
�� 577
�� 22. �� 578
22.1. �� 579
22.2. �� 580
22.3. �� 581
22.4. �� 583
22.5. �� 583
22.6. �� 587
22.7. �� 588
22.8. �� 589
22.9. �� 589
22.10. �� 590
22.11. �� 591
22.12. �� 591
22.13. ��, �� 592
22.14. �� 593
22.15. �� 593
22.16. �� 593
22.17. �� 594
�� 595
22.18. �� 595
22.19. �� 597
22.20. �� 598
�� 22.1. �� 599
�� 22.2. �� n(�) � �� ^n �� m(�) 599
�� 22.3. �� Tn(�) � �� ^n �� Tm(�) 600
�� 22.4. �� n(�) 600
�� 22.5. �� Un(x) � �� x^n �� Um(x) 601
�� 22.6. �� Un(x) 601
�� 22.7. �� Un(�) � �� x^n �� Um(�) 602
�� 22.8. �� Sn(x) � �� x^n �� Sm(x) 602
�� 22.9. �� n(�) � �� x^n �� m(�) 603
�� 22.10. �� Ln(x) � �� x^n �� Lm(x) 604
�� 22.11. �� Ln(x) 605
�� 22.12. �� Hn(�) � �� x^n �� Hm(�) 605
�� 22.13. �� Hn(�) 605
�� 606
�� 23. �� , �� , ��-�� 607
23.1. �� , �� 607
23.2. ��-�� 610
�� 23.1. �� 612
�� 23.2. �� 613
�� 23.3. �� 614
�� 23.4. �� 616
�� 23.5. x^n/n! 621
�� 24. �� 624
24.1. �� 625
24.1.1. �� 625
24.1.2. �� 625
24.1.3. �� 626
24.1.4. �� 627
24.2 �� 628
24.2.1. �� 628
24.2.2. �� 628
24.3. ��-�� 629
24.3.1. �� M�� 629
24.3.2 �� 629
24.3.3.�� σk(n) 629
24.3.4. �� 630
�� 24.1. �� 631
�� 24.2. �� 634
�� 24.3. �� 635
�� 24.4. �� 637
�� 24.5. �� 638
�� 24.6. �� 642
�� 24.7. �� 646
�� 24.8. �� , �� p-1 666
�� 672
�� 25. ��, �� 673
25.1. �� 674
25.2. �� 675
25.3. �� 679
25.4. �� 682
25.5. �� 692
�� 25.1. �� n �� 694
�� 25.2. �� k-�� n �� 708
�� 25.3. �� n �� 709
�� 25.4. �� 710
�� 25.5. �� 714
�� 25.6. �� 714
�� 25.7. �� 714
�� 25.8. �� , �� x^k 715
�� 25.9. �� 717
�� 25.10. �� 718
�� 25.11. �� 718
�� 719
�� 26. �� 721
26.1. �� ; �� 722
26.2. ��, �� , �� 728
26.3. �� 732
26.4. �� -�� 735
26.5. �� -�� 738
26.6. F-�� 740
26.7. t-�� 742
26.8. �� 743
26.9. �� 747
�� 26.1. �� 754
�� 26.2. �� log Q(x) �� 760
�� 26.3. �� 762
�� 26.4. �� Ζ(x) �� (�) � Q(x) 763
�� 26.5. �� (�) � Q(x) 764
�� 26.6. �� (�) � Q(x) 765
�� 26.7. �� χ^2, �� -��, �� 766
�� 26.8. �� χ^2-��; �� χ^2 �� Q � ν 772
�� 26.9. �� F-��; �� F �� Q, v1, v2 774
�� 26.10. �� t-��; �� t �� ν 778
�� 26.11. 2500 �� 779
�� 784
�� 27. �� 787
27.1. �� 788
27.2. �� 789
27.3. �� 789
27.4. �� 790
27.5. fm(x) � �� 791
27.7. �� 794
27.8. �� 795
27.9. �� 796
�� 28. �� 800
�� 800
�� 801
�� 28.1. 2^(�n) � �� 804
�� 28.2. 2^� � �� 805
�� 28.3. 10^(�n) � �� 805
�� 28.4. n lg2, n log_2 10 � �� 805
�� 28.5. �� 805
�� 28.6. �� 805
�� 806
�� 29. �� 807
29.1. �� 807
29.2. �� 807
29.3. �� 809
29.4. �� 815
�� 816
�� 826
�� 827