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