Handbook of Mathematics

Abramowitz and Stegun: Handbook of Mathematical Functions

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Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, by Milton Abramowitz and Irene A. Stegun Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, by Milton Abramowitz and Irene A. Stegun
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The present Handbook has been designed to provide scientific investigators and students with a comprehensive and self-contained summary of the mathematical functions that arise in physical and engineering problems.

Numerical tables of mathematical functions are in continual demand by scientists and engineers. A greater variety of functions and higher accuracy of tabulation are now required as a result of scientific advances and, especially, of the increasing use of automatic computers. In the latter connection, the tables serve mainly for preliminary surveys of problems before programming for machine operation. For those without easy access to machines, such tables are, of course, indispensable. The primary aim has been to include a maximum of useful information within the limits of a moderately large volume, with particular attention to the needs of scientists in all fields. An attempt has been made to cover the entire field of special functions. To carry out the goal set forth by the Ad Hoc Committee, it has been necessary to supplement the tables by including the mathematical properties that are important in computation work, as well as by providing numerical methods which demonstrate the use and extension of the tables.

The Handbook was prepared under the direction of the late Milton Abramowitz, and Irene A. Stegun.

For sponsorship opportunities please contact me.

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Note: pages omitted from this electronic edition are not hyperlinked.
Electronic release notes 0
Page index (hyperlinked) 0
Subject index (hyperlinked) 0
Title page I (HTML)
Errata Notice II (HTML)
Preface III (HTML)
Preface to the Ninth Printing IIIa (HTML)
Foreword V (HTML)
VI
Table of Contents VII (HTML)
VIII
Introduction. 1. Introduction. 2. Accuracy of the Tables. IX (HTML)
3. Auxiliary Functions and Arguments. 4. Interpolation X
XI
5. Inverse Interpolation XII
6. Bivariate Interpolation. 7. Generation of Functions from Recurrence Relations XIII
8. Acknowledgments XIV (HTML)
2. Physical Constants and Conversion Factors 5
Table 2.1. Common Units and Conversion Factors. 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. Table 2.5. Conversion Factors for Customary U.S. Units to Metric Units. Table 2.6. Geodetic Constants 8
3. Elementary analytical methods 9
3.1. Binomial Theorem and Binomial Coefficients; Arithmetic and Geometric Progressions; Arithmetic, Geometric, Harmonic and Generalized Means. 3.2. Inequalities 10
3.3. Rules for Differentiation and Integration 11
12
3.4. Limits, Maxima and Minima 13
3.5. Absolute and Relative Errors. 3.6. Infinite Series 14
15
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. 3.11. Use and Extension of the Tables. 3.12. Computing Techniques 19
20
References 23
4. Elementary Transcendental Functions: Logarithmic, Exponential, Circular and Hyperbolic Functions 65
Mathematical Properties. 4.1. Logarithmic Function 67
68
4.2. Exponential Function 69
70
4.3. Circular Functions 71
72
73
74
75
76
77
78
4.4. Inverse Circular Functions 79
80
81
82
4.5. Hyperbolic Functions 83
84
85
4.6. Inverse Hyperbolic Functions 86
87
88
Numerical Methods. 4.7. Use and Extension of the Tables 89
References 93
94
5. Exponential Integral and Related Functions 227
Mathematical Properties. 5.1. Exponential Integral 228
229
230
5.2. Sine and Cosine Integrals 231
232
Numerical Methods. 5.3. Use and Extension of the Tables 233
234
References 235
236
237
6. Gamma Function and Related Functions 253
Mathematical Properties. 6.1. Gamma Function 255
256
257
6.2. Beta Function. 6.3. Psi (Digamma) Function 258
259
6.4. Polygamma Functions. 6.5. Incomplete Gamma Function 260
261
262
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
266
7. Error Function and Fresnel Integrals 295
Mathematical Properties. 7.1. Error Function 297
298
7.2. Repeated Integrals of the Error Function 299
7.3. Fresnel Integrals 300
301
7.4. Definite and Indefinite Integrals 302
303
Numerical Methods. 7.5. Use and Extension of the Tables 304
References 308
309
Complex zeros, maxima, minima of the error function and Fresnel integrals: asymptotics 329
8. Legendre function 331
Mathematical Properties. Notation. 8.1. Differential Equation 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. 8.8. Integral Representations. 8.9. Summation Formulas. 8.10. Asymptotic Expansions 335
8.11. Toroidal Functions 336
8.12. Conical Functions. 8.13. Relation to Elliptic Integrals. 8.14. Integrals 337
338
Numerical Methods. 8.15. Use and Extension of the Tables 339
References 340
341
9. Bessel Functions of Integer Order 355
Mathematical Properties. Notation. Bessel Functions J and Y. 9.1. Definitions and Elementary Properties 358
359
360
361
362
363
9.2. Asymptotic Expansions for Large Arguments 364
9.3. Asymptotic Expansions for Large Orders 365
366
367
368
9.4. Polynomial Approximations 369
9.5. Zeros 370
371
372
373
Modified Bessel Functions I and K. 9.6. Definitions and Properties 374
375
376
9.7. Asymptotic Expansions 377
9.8. Polynomial Approximations 378
Kelvin Functions. 9.9. Definitions and Properties 379
380
9.10. Asymptotic Expansions 381
382
383
9.11. Polynomial Approximations 384
Numerical Methods. 9.12. Use and Extension of the Tables 385
386
387
References 388
389
10. Bessel Functions of Fractional Order 435
Mathematical Properties. 10.1. Spherical Bessel Functions 437
438
439
440
441
10.2. Modified Spherical Bessel Functions 443
444
10.3. Riccati-Bessel Functions 445
10.4. Airy Functions 446
447
448
449
450
451
Numerical Methods. 10.5. Use and Extension of the Tables 452
References 455
456
11. Integrals of Bessel Functions 479
Mathematical Properties. 11.1. Simple Integrals of Bessel Functions 480
481
11.2. Repeated Integrals of Jn(z) and K0(z) 482
11.3. Reduction Formulas for Indefinite Integrals 483
484
11.4. Definite Integrals 485
486
487
Numerical Methods. 11.5. Use and Extension of the Tables 488
489
References 490
491
12. Struve Functions and Related Functions 495
Mathematical Properties. 12.1. Struve Function Hn(s) 496
497
12.2. Modified Struve Function Lnu(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
13. Confluent Hypergeometric Functions 503
Mathematical Properties. 13.1. Definitions of Kummer and Whittaker Functions 504
13.2. Integral Representations 505
13.3. Connections With Bessel Functions 506
507
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
515
14. Coulomb Wave Functions 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
541
14.6. Special Values and Asymptotic Behavior 542
Numerical Methods. 14.7. Use and Extension of the Tables 543
References 544
15. Hypergeometric Functions 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
559
560
15.4. Special Cases of F(a, b; c; z), Polynomials and Legendre Functions 561
15.5. The Hypergeometric Differential Equation 562
563
15.6. Riemann's Differential Equation 564
15.7. Asymptotic Expansions. References 565
566
16. Jacobian Elliptic Functions and Theta Functions 567
568
Mathematical Properties. 16.1. Introduction 569
16.2. Classification of the Twelve Jacobian Elliptic Functions. 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.). 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. 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
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
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
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
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
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
16.37. Expression as Infinite Products. 16.38. Expression as Infinite Series. Numerical Methods. 16.39. Use and Extension of the Tables 579
References 581
17. Elliptic Integrals 587
Mathematical Properties. 17.1. Definition of Elliptic Integrals. 17.2. Canonical Forms 589
17.3. Complete Elliptic Integrals of the First and Second Kinds 590
591
17.4. Incomplete Elliptic Integrals of the First and Second Kinds 592
593
594
595
596
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
601
References 606
607
18. Weierstrass Elliptic and Related Functions 627
Mathematical Properties. 18.1. Definitions, Symbolism, Restrictions and Conventions 629
630
18.2. Homogeneity Relations, Reduction Formulas and Processes 631
632
18.3. Special Values and Relations 633
634
18.4. Addition and Multiplication Formulas. 18.5. Series Expansions 635
636
637
638
639
18.6. Derivatives and Differential Equations 640
18.7. Integrals 641
18.8. Conformal Mapping 642
643
644
645
646
647
648
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 (g2=0, g3=1) 652
653
654
655
656
657
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
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
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
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
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
20.3. Floquet's Theorem and Its Consequences 727
728
729
20.4. Other Solutions of Mathieu's Equation 730
731
20.5. Properties of Orthogonality and Normalization. 20.6. Solutions of Mathieu's Modified Equation for Integral nu 732
733
734
20.7. Representations by Integrals and Some Integral Equations 735
736
737
20.8. Other Properties 738
739
20.9. Asymptotic Representations 740
741
742
743
20.10. Comparative Notations 744
References 745
746
21. Spheroidal Wave Functions 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
754
755
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
22. Orthogonal Polynomials 771
Mathematical Properties. 22.1. Definition of Orthogonal Polynomials 773
22.2. Orthogonality Relations 774
22.3. Explicit Expressions 775
776
22.4. Special Values. 22.5. Interrelations 777
778
779
780
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
789
22.19. Least Square Approximations 790
22.20. Economization of Series 791
References 792
23. Bernoulli and Euler Polynomials, Riemann Zeta Function 803
Mathematical Properties. 23.1. Bernoulli and Euler Polynomials and the Euler-Maclaurin Formula 804
805
806
23.2. Riemann Zeta Function and Other Sums of Reciprocal Powers 807
References 808
24. Combinatorial Analysis 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
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

 

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