M. Abramowitz and I. A. Stegun.
Table of contents ::
Index to all pages and sections ::
Subject index
Go straight to chapter …
Preface …… Page III
Foreword …… Page V
Contents …… Page VII
Introduction …… Page IX
1. Mathematical Constants …… Page 1
2. Physical Constants and Conversion Factors …… Page 5
3. Elementary Analytical Methods …… Page 9
4. Elementary Transcendental FunctionsLogarithmic, Exponential, Circular and Hyperbolic Functions …… Page 65
5. Exponential Integral and Related Functions …… Page 227
6. Gamma Function and Related Functions …… Page 253
7. Error Function and Fresnel Integrals …… Page 295
8. Legendre Functions …… Page 331
9. Bessel Functions of Integer Order …… Page 355
10. Bessel Functions of Fractional Order …… Page 435
11. Integrals of Bessel Functions …… Page 479
12. Struve Functions and Related Functions …… Page 495
13. Confluent Hypergeometric Functions …… Page 503
14. Coulomb Wave Functions …… Page 537
15. Hypergeometric Functions …… Page 555
16. Jacobian Elliptic Functions and Theta Functions …… Page 567
17. Elliptic Integrals …… Page 587
18. Weierstrass Elliptic and Related Functions …… Page 627
19. Parabolic Cylinder Functions …… Page 685
20. Mathieu Functions …… Page 721
21. Spheroidal Wave Functions …… Page 751
22. Orthogonal Polynomials …… Page 771
23. Bernoulli and Euler Polynomials, Riemann Zeta Function …… Page 803
24. Combinatorial Analysis …… Page 821
25. Numerical Interpolation, Differentiation and Integration …… Page 875
26. Probability Functions …… Page 925
27. Miscellaneous Functions …… Page 997
28. Scales of Notation …… Page 1011
29. Laplace Transforms …… Page 1019
Subject Index …… Page 1031
Index of Notations …… Page 1044
Index to all pages and sections
1. Introduction IX [HTML ]
2. Accuracy of the Tables IX [HTML ]
3. Auxiliary Functions and Arguments X
4. Interpolation X
5. Inverse Interpolation XII
6. Bivariate Interpolation XIII
7. Generation of Functions from Recurrence Relations XIII
8. Acknowledgments XIV [HTML ]
By M. G. McNish
Contents 5
Table 2.1. Common Units and Conversion Factors. 6
Table 2.2. Names and Conversion Factors for Electric and Magnetic Units 6
Table 2.3. Adjusted Values of Constants 7
Table 2.4. Miscellaneous Conversion Factors 8
Table 2.5. Conversion Factors for Customary U.S. Units to Metric Units 8
Table 2.6. Geodetic Constants 8
By Milton Abramowitz
Contents 9
3.1. Binomial Theorem and Binomial Coefficients; Arithmetic and Geometric Progressions; 10
Arithmetic, Geometric, Harmonic and Generalized Means 10
3.2. Inequalities 10
3.3. Rules for Differentiation and Integration 11
3.4. Limits, Maxima and Minima 13
3.5. Absolute and Relative Errors. 3.6. Infinite Series 14
3.7. Complex Numbers and Functions 16
3.8. Algebraic Equations 17
3.9. Successive Approximation Methods 18
3.10. Theorems on Continued Fractions. Numerical Methods 19
3.11. Use and Extension of the Tables 19
3.12. Computing Techniques 19
References 23
By Ruth Zucker
Contents 65
Mathematical Properties. 4.1. Logarithmic Function 67
4.2. Exponential Function 69
4.3. Circular Functions 71
4.4. Inverse Circular Functions 79
4.5. Hyperbolic Functions 83
4.6. Inverse Hyperbolic Functions 86
Numerical Methods. 4.7. Use and Extension of the Tables 89
References 93
By Walter Gautschi and William F. Cahill
Contents 227
5.1. Exponential Integral - Mathematical Properties 228
5.2. Sine and Cosine Integrals 231
5.3. Use and Extension of the Tables - Numerical Methods 233
References 235
By Philip J. Davis
Contents 253
Mathematical Properties. 6.1. Gamma Function 255
6.2. Beta Function. 6.3. Psi (Digamma) Function 258
6.4. Polygamma Functions. 6.5. Incomplete Gamma Function 260
6.6. Incomplete Beta Function. Numerical Methods. 6.7. Use and Extension of the Tables 263
6.8. Summation of Rational Series by Means of Polygamma Functions 264
References 265
By Walter Gautschi
Contents 295
Mathematical Properties. 7.1. Error Function 297
7.2. Repeated Integrals of the Error Function 299
7.3. Fresnel Integrals 300
7.4. Definite and Indefinite Integrals 302
Numerical Methods. 7.5. Use and Extension of the Tables 304
References 308
Complex zeros, maxima, minima of the error function and Fresnel integrals: asymptotics 329
By Irene A. Stegun
Contents 331
8.1. Differential Equation - Mathematical Properties. Notation 332
8.2. Relations Between Legendre Functions. 8.3. Values on the Cut. 8.4. Explicit Expressions 333
8.6. Special Values 334
8.7. Trigonometric Expansions 335
8.8. Integral Representations 335
8.9. Summation Formulas 335
8.10. Asymptotic Expansions 335
8.11. Toroidal Functions 336
8.12. Conical Functions 337
8.13. Relation to Elliptic Integrals 337
8.14. Integrals 337
8.15. Use and Extension of the Tables - Numerical Methods 339
References 340
By F. W. J. Olver
Contents 355
9.1. Definitions and Elementary Properties - Mathematical Properties. Notation. Bessel Functions J and Y 358
9.2. Asymptotic Expansions for Large Arguments 364
9.3. Asymptotic Expansions for Large Orders 365
9.4. Polynomial Approximations 369
9.5. Zeros 370
Modified Bessel Functions I and K . 9.6. Definitions and Properties 374
9.7. Asymptotic Expansions 377
9.8. Polynomial Approximations 378
Kelvin Functions. 9.9. Definitions and Properties 379
9.10. Asymptotic Expansions 381
9.11. Polynomial Approximations 384
Numerical Methods. 9.12. Use and Extension of the Tables 385
References 388
By H. A. Antosiewicz
Contents 435
Mathematical Properties. 10.1. Spherical Bessel Functions 437
10.2. Modified Spherical Bessel Functions 443
10.3. Riccati-Bessel Functions 445
10.4. Airy Functions 446
Numerical Methods. 10.5. Use and Extension of the Tables 452
References 455
By Yudell L. Luke
Contents 479
Mathematical Properties. 11.1. Simple Integrals of Bessel Functions 480
11.2. Repeated Integrals of Jn (z ) and K 0 (z ) 482
11.3. Reduction Formulas for Indefinite Integrals 483
11.4. Definite Integrals 485
Numerical Methods. 11.5. Use and Extension of the Tables 488
References 490
By Milton Abramowitz
Contents 495
Mathematical Properties. 12.1. Struve Function H n (s ) 496
12.2. Modified Struve Function L nu (z ). 12.3. Anger and Weber Functions 498
Numerical Methods. 12.4. Use and Extension of the Tables 499
References 500
Explanations of numerical methods to compute Struve functions 502
By Lucy Joan Slater
Contents 503
Mathematical Properties. 13.1. Definitions of Kummer and Whittaker Functions 504
13.2. Integral Representations 505
13.3. Connections With Bessel Functions 506
13.5. Asymptotic Expansions and Limiting Forms 508
13.6. Special Cases 509
13.7. Zeros and Turning Values 510
Numerical Methods. 13.8. Use and Extension of the Tables 511
13.10. Graphing M (a , b , x ) 513
References 514
By Milton Abramowitz
Contents 537
Mathematical Properties. 14.1. Differential Equation, Series Expansions 538
14.2. Recurrence and Wronskian Relations. 14.3. Integral Representations. 14.4. Bessel Function Expansions 539
14.5. Asymptotic Expansions 540
14.6. Special Values and Asymptotic Behavior 542
Numerical Methods. 14.7. Use and Extension of the Tables 543
References 544
By Fritz Oberhettinger
Contents 555
Mathematical Properties. 15.1. Gauss Series, Special Elementary Cases, Special Values of the Argument 556
15.2. Differentiation Formulas and Gauss' Relations for Contiguous Functions 557
Integral Representations and Transformation Formulas 558
15.4. Special Cases of F (a , b ; c ; z ), Polynomials and Legendre Functions 561
15.5. The Hypergeometric Differential Equation 562
15.6. Riemann's Differential Equation 564
15.7. Asymptotic Expansions. References 565
By L. M. Milne-Thomson
Contents 567
Mathematical Properties. 16.1. Introduction 569
16.2. Classification of the Twelve Jacobian Elliptic Functions 570
16.3. Relation of the Jacobian Functions to the Copolar Trio 570
16.4. Calculation of the Jacobian Functions by Use of the Arithmetic-Geometric Mean (A.G.M.) 571
16.5. Special Arguments. 16.6. Jacobian Functions when m =0 or 1 571
16.7. Principal Terms. 16.8. Change of Argument 572
16.9. Relations Between the Squares of the Functions. 16.10. Change of Parameter 573
16.11. Reciprocal Parameter (Jacobi's Real Transformation) 573
16.12. Descending Landen Transformation (Gauss' Transformation) 573
16.14. Ascending Landen Transformation 573
16.15. Approximation in Terms of Hyperbolic Functions 574
16.16. Derivatives. 16.17. Addition Theorems 574
16.18. Double Arguments. 16.19. Half Arguments 574
16.20. Jacobi's Imaginary Transformation 574
16.21. Complex Arguments. 16.22. Leading Terms of the Series in Ascending Powers of u 575
16.23. Series Expansion in Terms of the Nome q and the Argument v 575
16.24. Integrals of the Twelve Jacobian Elliptic Functions 575
16.25. Notation for the Integrals of the Squares of the Twelve Jacobian Elliptic Functions 576
16.26. Integrals in Terms of the Elliptic Integral of the Second Kind 576
16.27. Theta Functions; Expansions in Terms of the Nome q 576
16.28. Relations Between the Squares of the Theta Functions 576
16.29. Logarithmic Derivatives of the Theta Functions 576
16.30. Logarithms of Theta Functions of Sum and Difference 577
16.31. Jacobi's Notation for Theta Functions 577
16.32. Calculation of Jacobi's Theta Function Theta(u |m ) by Use of the Arithmetic-Geometric Mean 577
16.33. Addition of Quarter-Periods to Jacobins Eta and Theta Functions 577
16.34. Relation of Jacobi's Zeta Function to the Theta Functions 578
16.35. Calculation of Jacobi's Zeta Function Z (u |m ) by Use of the Arithmetic-Geometric Mean 578
16.36. Neville's Notation for Theta Functions 578
16.37. Expression as Infinite Products 579
16.38. Expression as Infinite Series 579
Numerical Methods. 16.39. Use and Extension of the Tables 579
References 581
By L. M. Milne-Thomson
Contents 587
Mathematical Properties. 17.1. Definition of Elliptic Integrals 589
17.2. Canonical Forms 589
17.3. Complete Elliptic Integrals of the First and Second Kinds 590
17.4. Incomplete Elliptic Integrals of the First and Second Kinds 592
17.5. Landen's Transformation 597
17.6. The Process of the Arithmetic-Geometric Mean 598
17.7. Elliptic Integrals of the Third Kind 599
Numerical Methods. 17.8. Use and Extension of the Tables 600
References 606
By Thomas H. Southard
Contents 627
Mathematical Properties. 18.1. Definitions, Symbolism, Restrictions and Conventions 629
18.2. Homogeneity Relations, Reduction Formulas and Processes 631
18.3. Special Values and Relations 633
18.4. Addition and Multiplication Formulas. 18.5. Series Expansions 635
18.6. Derivatives and Differential Equations 640
18.7. Integrals 641
18.8. Conformal Mapping 642
18.9. Relations with Complete Elliptic Integrals K and K' and Their Parameter m and with Jacobins Elliptic Functions 649
18.10. Relations with Theta Functions 650
18.11. Expressing any Elliptic Function in Terms of P and P' 651
18.13. Equianharmonic Case (g 2 =0, g 3 =1) 652
18.14. Lemniscatic Case (g 2 =1, g 3 =0) 658
18.15. Pseudo-Lemniscatic Case (g 2 =-1, g 3 =0) 662
Numerical Methods. 18.16. Use and Extension of the Tables 663
References 670
By J. C. P. Miller
Contents 685
Mathematical Properties. 19.1. The Parabolic Cylinder Functions, Introductory 686
The Equation d 2 y /dx 2 -(x 2 /4+a )y =0. 19.2 to 19.6. Power Series, Standard Solutions, Wronskian and Other Relations, Integral Representations, Recurrence Relations 686
19.7 to 19.11. Asymptotic Expansions 689
19.12 to 19.15. Connections With Other Functions 691
The Equation d 2 y /dx 2 +(x 2 /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
19.25. Connections With Other Functions 695
19.26. Zeros 696
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
References 700
By Gertrude Blanch
Contents 721
Mathematical Properties. 20.1. Mathieu's Equation. 20.2. Determination of Characteristic Values 722
20.3. Floquet's Theorem and Its Consequences 727
20.4. Other Solutions of Mathieu's Equation 730
20.5. Properties of Orthogonality and Normalization. 20.6. Solutions of Mathieu's Modified Equation for Integral nu 732
20.7. Representations by Integrals and Some Integral Equations 735
20.8. Other Properties 738
20.9. Asymptotic Representations 740
20.10. Comparative Notations 744
References 745
By Arnold N. Lowan
Contents 751
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
21.6. Differential Equations for Radial and Angular Spheroidal Wave Functions. 21.7. Prolate Angular Functions 753
21.8. Oblate Angular Functions. 21.9. Radial Spheroidal Wave Functions 756
21.10. Joining Factors for Prolate Spheroidal Wave Functions 757
21.11. Notation 758
References 759
By Urs W. Hochstrasser
Contents 771
Mathematical Properties. 22.1. Definition of Orthogonal Polynomials 773
22.2. Orthogonality Relations 774
22.3. Explicit Expressions 775
22.4. Special Values. 22.5. Interrelations 777
22.6. Differential Equations 781
22.7. Recurrence Relations 782
22.8. Differential Relations. 22.9. Generating Functions 783
22.10. Integral Representations 784
22.11. Rodrigues' Formula. 22.12. Sum Formulas. 22.13. Integrals Involving Orthogonal Polynomials 785
22.14. Inequalities 786
22.15. Limit Relations. 22.16. Zeros 787
22.17. Orthogonal Polynomials of a Discrete Variable. Numerical Methods. 22.18. Use and Extension of the Tables 788
22.19. Least Square Approximations 790
22.20. Economization of Series791
References 792
By Emilie V. Haynsworth and Karl Goldberg
Contents 803
Mathematical Properties. 23.1. Bernoulli and Euler Polynomials and the Euler-Maclaurin Formula 804
23.2. Riemann Zeta Function and Other Sums of Reciprocal Powers 807
References 808
By K. Goldberg, M. Newman and E. Haynsworth
Contents 821
Mathematical Properties. 24.1. Basic Numbers. 24.1.1. Binomial Coefficients 822
24.1.2. Multinomial Coefficients 823
24.1.3. Stirling Numbers of the First Kind. 24.1.4. Stirling Numbers of the Second Kind 824
24.2. Partitions. 24.2.1. Unrestricted Partitions. 24.2.2. Partitions Into Distinct Parts 825
24.3. Number Theoretic Functions. 24.3.1. The Mobius Function. 24.3.2. The Euler Function 826
24.3.3. Divisor Functions. 24.3.4. Primitive Roots. References 827
By Philip J. Davis and Ivan Polonsky
Contents 875
25.1. Differences 877
25.2. Interpolation 878
25.3. Differentiation 882
25.4. Integration 885
25.5. Ordinary Differential Equations 896
References 898
By Marvin Zelen and Norman C. Severo
Contents 925
Mathematical Properties. 26.1. Probability Functions: Definitions and Properties 927
26.2. Normal or Gaussian Probability Function 931
26.3. Bivariate Normal Probability Function 936
26.4. Chi-Square Probability Function 940
26.5. Incomplete Beta Function 944
26.6. F -(Variance-Ratio) Distribution Function 946
26.7. Student's t -Distribution 948
Numerical Methods. 26.8. Methods of Generating Random Numbers and Their Applications 949
26.9. Use and Extension of the Tables 953
References 961
By Irene A. Stegun
Contents 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
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
By S. Peavy and A. Schopp
Contents 1011
Representation of numbers 1012
Numerical methods 1013
References 1015
Table 28.1. 2±n in Decimal, n =0(1)50, Exact 1016
Table 28.2. 2x in Decimal, x =.001(.001).01(.01).1(.1).9, 15D 1017
Table 28.3. 10±n in Octal, n =0(1)18, Exact or 20D 1017
Table 28.4. n log10 2, n log2 10 in Decimal, n =1(1)10, 10D 1017
Table 28.5. Addition and Multiplication Tables, Binary and Octal Scales 1017
Table 28.6. Mathematical Constants in Octal Scale 1017
Contents 1019
29.1. Definition of the Laplace Transform. 29.2. Operations for the Laplace Transform 1020
29.3. Table of Laplace Transforms 1021
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
Notation - Greek Letters. Miscellaneous Notations 1046
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Last modified Wednesday 15 Feb 2023