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Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions with formulas, graphs, and mathematical tables

Published $1964$, Dover Publications

ISBN 0-486-61272-4.

Online mirrors

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BiBTeX

@book {MR0167642,
    AUTHOR = {Abramowitz, Milton and Stegun, Irene A.},
     TITLE = {Handbook of mathematical functions with formulas, graphs, and
              mathematical tables},
    SERIES = {National Bureau of Standards Applied Mathematics Series},
    VOLUME = {55},
 PUBLISHER = {For sale by the Superintendent of Documents, U.S. Government
              Printing Office, Washington, D.C.},
      YEAR = {1964},
     PAGES = {xiv+1046},
}

Contents

Preface

Foreword

Introduction

1. Mathematical Constants

2. Physical Constants and Conversion Factors

3. Elementary Analytical Methods

3.1. Binomial Theorem and Binomial Coefficients; Arithmetic and Geometric Progressions; Arithmetic, Geometric, Harmonic and Generalized Means

3.2. Inequalities

3.3. Rules for Differentiation and Integration

$3.3.1$

$3.3.2$

$3.3.3$

$3.3.4$

$3.3.5$

3.4. Limits, Maxima and Minima

3.5. Absolute and Relative Errors

3.6. Infinite Series

3.7. Complex Numbers and Functions

3.8. Algebraic Equations

3.9. Successive Approximation Methods

3.10. Theorems on Continued Fractions

3.11. Use and Extension of the Tables

3.12. Computing Techniques

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

4.1. Logarithmic Function

$4.1.1$

$4.1.2$

$4.1.3$

$4.1.4$

$4.1.5$

$4.1.6$

$4.1.7$

$4.1.8$

$4.1.9$

$4.1.10$

$4.1.11$

$4.1.12$

$4.1.13$

$4.1.14$

$4.1.15$ (also $4.1.15$)

$4.1.16$

$4.1.17$

$4.2.18$

$4.2.19$

$4.2.20$

$4.2.21$

$4.2.22$

$4.2.23$

$4.1.24$

$4.1.25$

$4.1.26$

$4.1.27$

$4.1.28$

$4.1.29$

4.2. Exponential Function

$4.2.1$

$4.2.2$

$4.2.3$

$4.2.4$

$4.2.5$

$4.2.6$

$4.2.7$

$4.2.8$

$4.2.9$

$4.2.10$

$4.2.11$

$4.2.12$

$4.2.13$

$4.2.29$

$4.2.30$

$4.2.31$

$4.2.32$

$4.2.33$

$4.2.34$

$4.2.35$

$4.2.36$

$4.2.37$

$4.2.38$

$4.2.39$

4.3. Circular Functions

$4.3.1$

$4.3.2$

$4.3.3$

$4.3.4$

$4.3.5$

4.3.6

4.4. Inverse Circular Functions

4.5. Hyperbolic Functions

$4.5.1$

$4.5.2$

$4.5.3$

$4.5.4$

$4.5.5$

$4.5.6$

$4.5.7$

$4.5.8$

$4.5.9$

$4.5.10$

$4.5.11$

$4.5.12$

4.6. Inverse Hyperbolic Functions

4.7. Use and Extension of the Tables

5. Exponential Integral and Related Functions

5.1. Exponential Integral

5.2. Sine and Cosine Integrals

5.3. Use and Extension of the Tables

6. Gamma Function and Related Functions

6.1. Gamma Function

$6.1.1$

$6.1.2$

$6.1.3$

$6.1.4$

$6.1.5$

$6.1.6$

$6.1.7$

$6.1.8$

$6.1.9$

$6.1.10$

$6.1.11$

$6.1.12$

$6.1.13$

$6.1.14$

$6.1.15$

$6.1.16$

$6.1.17$

$6.1.18$

$6.1.19$

$6.1.20$

$6.1.21$

$6.1.22$

$6.1.23$

$6.1.24$

$6.1.25$

$6.1.26$

$6.1.27$

$6.1.28$

$6.1.29$

$6.1.30$

$6.1.31$

$6.1.32$

$6.1.33$

$6.1.34$

$6.1.35$

$6.1.36$

$6.1.37$

$6.1.38$

$6.1.39$

$6.1.40$

$6.1.41$

$6.1.42$

$6.1.43$

$6.1.44$

$6.1.45$

$6.1.46$

$6.1.47$

$6.1.48$

$6.1.49$

$6.1.50$

6.2. Beta Function

6.3. Psi (Digamma Function)

6.4. Polygamma Functions

$6.4.1$ (and $6.4.1$)

$6.4.2$

$6.4.3$

$6.4.4$

$6.4.5$

$6.4.6$

$6.4.7$

$6.4.8$

$6.4.9$

$6.4.10$

$6.4.11$

$6.4.12$

$6.4.13$

$6.4.14$

6.5. Incomplete Gamma Function

6.6. Incomplete Beta Function

6.7. Use and Extension of the Tables

6.8. Summation of Rational Series by Means of Polygamma Functions

7. Error Function and Fresnel Integrals

7.1. Error Function

$7.1.1$

$7.1.2$

$7.1.3$

$7.1.4$

$7.1.5$

$7.1.6$

$7.1.7$

$7.1.8$

$7.1.9$

$7.1.10$

$7.1.11$

$7.1.12$

$7.1.13$

$7.1.14$

$7.1.15$

$7.1.16$

$7.1.17$

$7.1.18$

$7.1.19$

$7.1.20$

$7.1.21$

$7.1.22$

$7.1.23$

$7.1.24$

$7.1.25$

$7.1.26$

$7.1.27$

$7.1.28$

$7.1.29$

7.2. Repeated Integrals of the Error Function

7.3. Fresnel Integrals

7.4. Definite and Indefinite Integrals

7.5. Use and Extension of the Tables

8. Legendre Functions

8.1. Differential Equation

8.2. Relations Between Legendre Functions

8.3. Values on the Cut

8.4. Explicit Expressions

8.5. Recurrence Relations

8.6. Special Values

8.7. Trigonometric Expressions

8.8. Integral Representations

8.9. Summation Formulas

8.10. Asymptotic Expansions

8.11. Toroidal Functions

8.12. Conical Functions

8.13. Relation to Elliptic Integrals

8.14. Integrals

8.15. Use and Extension of the Tables

9. Bessel Functions of Integer Order

9.1. Definitions and Elementary Properties

$9.1.1$

$9.1.2$

$9.1.3$ (and $9.1.3$)

$9.1.4$ (and $9.1.4$)

9.1.5 (and $9.1.5$)

$9.1.6$

$9.1.7$

$9.1.8$

$9.1.9$

$9.1.10$

$9.1.11$

$9.1.12$

$9.1.13$

$9.1.14$

$9.1.15$

$9.1.16$

$9.1.17$

$9.1.18$

$9.1.19$

$9.1.20$

$9.1.21$

$9.1.22$

$9.1.23$

$9.1.24$

$9.1.25$

$9.1.26$

$9.1.27$

$9.1.28$

$9.1.29$

$9.1.30$

$9.1.31$

$9.1.32$

$9.1.33$

$9.1.34$

$9.1.35$

$9.1.36$

$9.1.37$

$9.1.38$

$9.1.39$

$9.1.40$

$9.1.41$

$9.1.42$

$9.1.43$

$9.1.44$

$9.1.45$

$9.1.46$

$9.1.47$

$9.1.48$

$9.1.49$

$9.1.50$

$9.1.51$

$9.1.52$

$9.1.53$

$9.1.54$

$9.1.55$

$9.1.56$

$9.1.57$

$9.1.58$

$9.1.59$

$9.1.60$

$9.1.61$

$9.1.62$

$9.1.64$

$9.1.65$

$9.1.66$

$9.1.67$

$9.1.68$

$9.1.69$

$9.1.70$

$9.1.71$

$9.1.72$

$9.1.73$

$9.1.74$

$9.1.75$

$9.1.76$

$9.1.77$

$9.1.78$

$9.1.79$

$9.1.80$

$9.1.81$

$9.1.82$

$9.1.83$

$9.1.84$

$9.1.85$

$9.1.86$

$9.1.87$

$9.1.88$

$9.1.89$

9.2. Asymptotic Expansions for Large Arguments

9.3. Asymptotic Expansions for Large Orders

9.4. Polynomial Approximations

9.5. Zeros

9.6. Definitions and Properties

9.7. Asymptotic Expansions

9.8. Polynomial Approximations

9.9. Definitions and Properties

9.10. Asymptotic Expansions

9.11. Polynomial Approximations

9.12. Use and Extension of the Tables

10. Bessel Functions of Fractional Order

10.1 Spherical Bessel Functions

10.2 Modified Spherical Bessel Functions

10.3 Riccati-Bessel Functions

10.4 Airy Functions

10.5 Use and Extension of the Tables

11. Integrals of Bessel Functions

11.1 Simple Integrals of Bessel Functions

$11.1.1$

$11.1.2$

$11.1.3$

$11.1.4$

$11.1.5$

$11.1.6$

$11.1.7$

$11.1.8$

$11.1.9$

$11.1.10$

$11.1.11$

$11.1.12$

$11.1.13$

$11.1.14$

$11.1.15$

$11.1.16$

$11.1.17$

$11.1.18$

$11.1.19$

$11.1.20$

$11.1.21$

$11.1.22$

$11.1.23$

$11.1.24$

$11.1.25$

$11.1.26$

$11.1.27$

$11.1.28$

$11.1.29$

$11.1.30$

$11.1.31$

11.2 Repeated Integrals of $J_n(z)$ and $K_0(z)$

11.3 Reduction Formulas for Indefinite Integrals

11.4 Definite Integrals

11.5 Use and Extensions of the Tables

12. Struve Functions and Related Functions

12.1 Struve Function $\mathbf{H}_{\nu}(z)$

12.2 Modified Struve Functions $\mathbf{L}_{\nu}(z)$

12.3 Anger and Weber Functions

$12.3.1$

$12.3.2$

$12.3.3$

$12.3.4$

$12.3.5$

$12.3.6$

$12.3.7$

$12.3.8$

$12.3.9$

$12.3.10$

12.4 Use and Extension of the Tables

13. Confluent Hypergeometric Functions

13.1 Definitions of Kummer and Whittaker Functions

13.2 Integral Representations

13.3 Connections With Bessel Functions

13.4 Recurrence Relations and Differential Properties

13.5 Asymptotic Expansions and Limiting Forms

13.6 Special Cases

13.7 Zeros and Turning Values

13.8 Use and Extension of the Tables

13.9 Calculation of the Zeros and Turning Points

13.10 Graphing $M(a,b,x)$

14. Coulomb Wave Functions

14.1 Differential Equation, Series Expansions

14.2 Recurrence and Wronskian Relations

14.3 Integral Representations

14.4 Bessel Function Expansions

14.5 Asymptotic Expansions

14.6 Special Values and Asymptotic Behavior

14.7 Use and Extension of the Tables

15. Hypergeometric Functions

15.1 Gauss Series, Special Elementary Cases, Special Values of the Argument

15.1.1

15.1.2

15.1.3

15.1.4

15.1.5

15.1.6

15.1.7

15.1.8

15.1.9

15.1.10

15.1.11

15.1.12

15.1.13

15.1.14

15.1.15

15.1.16

15.1.17

15.1.18

15.1.19

15.1.20

15.1.21

15.1.22

15.1.23

15.1.24

15.1.25

15.1.26

15.1.27

15.1.28

15.1.29

15.1.30

15.1.31

15.2 Differentiation Formulas and Gauss' Relations for Contiguous Functions

15.3 Integral Representations and Transformation Formulas

15.4 Special Cases of $F(a,b;c;z)$, Polynomials and Legendre Functions

15.5 The Hypergeometric Differential Equation

15.6 Riemann's Differential Equation

15.7 Asymptotic Expansions

16. Jacobian Elliptic Functions and Theta Functions

16.1 Introduction

16.2 Classification of the Twelve Jacobian Elliptic Functions

16.3 Relation of the Jacobian Functions to the Copolor Trio

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 $1$

16.7 Principal Terms

16.8 Change of Argument

16.9 Relations Between the Squares of the Functions

16.10 Change of Parameter

16.11 Reciprocal Parameter

16.12 Descending Landen Transformation (Gauss' Transformation)

16.13 Approximation in Terms of Circular Functions

16.14 Ascending Landen Transformation

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

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 $q$

16.24 Integrals of the Twelve Jacobian Elliptic Functions

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 $q$

16.27.1

16.27.2

16.27.3

16.27.4

16.28 Relations Between the Squares of the Theta Functions

16.28.1

16.28.2

16.28.3

16.28.4

16.28.5

16.28.6

16.28.5

16.28.6

16.29 Logarithmic Derivatives of the Theta Functions

16.29.1

16.29.2

16.29.3

16.29.4

16.30 Logarithms of Theta Functions of Sum and Difference

16.30.1

16.30.2

16.30.3

16.30.4

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 Jacobi's Eta and Theta Functions

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

16.37 Expression as Infinite Products

16.38 Expression as Infinite Series

16.39 Use and Extension of the Tables

17. Elliptic Integrals

17.1 Definition of Elliptic Integrals

17.2 Canonical Forms

17.3 Complete Elliptic Integrals of the First and Second Kinds

17.4 Incomplete Elliptic Integrals of the First and Second Kinds

17.5 Landen's Transformation

17.6 The Process of the Arithmetic-Geometric Mean

17.7 Elliptic Integrals of the Third Kind

17.8 Use and Extension of the Tables

18. Weierstrass Elliptic and Related Functions

18.1 Definitions, Symbolism, Restrictions and Conventions

18.2 Homogeneity Relations, Reduction Formulas and Processes

18.3 Special Values and Relations

18.4 Addition and Multiplication Formulas

18.5 Series Expansions

18.6 Derivatives and Differential Equations

18.7 Integrals

18.8 Conformal Mapping

18.9 Relations with Complete Elliptic Integrals $K$ and $K'$ and Their Parameter $m$ and with Jacobi's Elliptic Functions

18.10 Relations with Theta Functions

18.11 Expressing and Elliptic Function in Terms of $\wp$ and $\wp'$

18.12 Case $\Delta=0$

18.13 Equianharmonic Case ($g_2=0,g_3=1$)

18.14 Lemniscatic Case ($g_2=1, g_3=0$)

18.15 Pseudo-Lemniscatic Case ($g_2=-1, g_3=0$)

18.16 Use and Extension of the Tables

19. Parabolic Cylinder Functions

20. Mathieu Functions

21. Spheroidal Wave Functions

22. Orthogonal Polynomials

23. Bernoulli and Euler Polynomials, Riemann Zeta Function

24. Combinatorial Analysis

24.1. Basic numbers

24.1.1. Binomial Coefficients

24.1.2. Multinomial Coefficients

24.1.3. Stirling Numbers of the First Kind

24.1.4. Stirling Numbers of the Second Kind

24.2. Partitions

24.2.1. Unrestricted Partitions

$24.2.1 \mathrm{I}.A.$

$24.2.1 \mathrm{I}.B.$

$24.2.1 \mathrm{II}.A.$

$24.2.1 \mathrm{II}.A.$

$24.2.1 \mathrm{II}.A.$

$24.2.1 \mathrm{II}.B.$

$24.2.1 \mathrm{III}$

24.2.2. Partitions Into Distinct Parts

24.3. Number Theoretic Functions

24.3.1. The Mobius Function

$24.3.1 \mathrm{I}.A.$

$24.3.1. \mathrm{I}.B.$

$24.3.1 \mathrm{I}.B.$

$24.3.1 \mathrm{II}.A.$

24.3.2. The Euler Function

$24.3.2. \mathrm{I}.A$

$24.3.2 \mathrm{I}.B.$

$24.3.2 \mathrm{I}.B.$

$24.3.2 \mathrm{I}.C.$

$24.3.2 \mathrm{II}.A.$

24.3.3. Divisor Functions

$24.3.3I.A.$

$24.3.3I.B.$

24.3.4. Primitive Roots

References

25. Numerical Interpolation, Differentiation and Integration

26. Probability Functions

27. Miscellaneous Functions

28. Scales of Notation

29. Laplace Transforms

Subject Index

Index of Notations