M. Abramowitz and I. A. Stegun.

Handbook of mathematical functions.

Table of contents :: Index to all pages and sections :: Subject index

Index to all pages and sections

1. Introduction

Top of Index

  • 1. IntroductionIX [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]

2. Physical Constants and Conversion Factors

Top of Index

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

3. Elementary analytical methods

Top of Index

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

4. Elementary Transcendental Functions: Logarithmic, Exponential, Circular and Hyperbolic Functions

Top of Index

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

5. Exponential Integral and Related Functions

Top of Index

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

6. Gamma Function and Related Functions

Top of Index

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

7. Error Function and Fresnel Integrals

Top of Index

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

8. Legendre function

Top of Index

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

9. Bessel Functions of Integer Order

Top of Index

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

10. Bessel Functions of Fractional Order

Top of Index

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

11. Integrals of Bessel Functions

Top of Index

By Yudell L. Luke

  • Contents 479
  • Mathematical Properties. 11.1. Simple Integrals of Bessel Functions 480
  • 11.2. Repeated Integrals of Jn(z) and K0(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

12. Struve Functions and Related Functions

Top of Index

By Milton Abramowitz

  • Contents 495
  • Mathematical Properties. 12.1. Struve Function Hn(s) 496
  • 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

Top of Index

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

14. Coulomb Wave Functions

Top of Index

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

15. Hypergeometric Functions

Top of Index

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

16. Jacobian Elliptic Functions and Theta Functions

Top of Index

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

17. Elliptic Integrals

Top of Index

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

18. Weierstrass Elliptic and Related Functions

Top of Index

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 (g2=0, g3=1) 652
  • 18.14. Lemniscatic Case (g2=1, g3=0) 658
  • 18.15. Pseudo-Lemniscatic Case (g2=-1, g3=0) 662
  • Numerical Methods. 18.16. Use and Extension of the Tables 663
  • References 670

19. Parabolic Cylinder Functions

Top of Index

By J. C. P. Miller

  • Contents 685
  • Mathematical Properties. 19.1. The Parabolic Cylinder Functions, Introductory 686
  • 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
  • 19.7 to 19.11. Asymptotic Expansions 689
  • 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
  • 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

20. Mathieu Functions

Top of Index

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

21. Spheroidal Wave Functions

Top of Index

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

22. Orthogonal Polynomials

Top of Index

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

23. Bernoulli and Euler Polynomials, Riemann Zeta Function

Top of Index

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

24. Combinatorial Analysis

Top of Index

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

25. Numerical Interpolation, Differentiation, and Integration

Top of Index

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

26. Probability Functions

Top of Index

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

27. Miscellaneous Functions

Top of Index

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

28. Scales of notation

Top of Index

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. nlog102, nlog210 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

29. Laplace Transforms

Top of Index

  • 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

Top of Index

  • 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

Top of Index

  • Index of Notations 1044
  • Notation - Greek Letters. Miscellaneous Notations 1046

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Last modified Wednesday 15 Feb 2023