# Book:Milton Abramowitz/Handbook of mathematical functions

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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.

### 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 $4.1.15$)
$4.1.16$
$4.1.17$
$4.1.18$
$4.1.19$
$4.1.20$
$4.1.21$
$4.1.22$ (and $4.1.22$)
$4.1.23$
$4.1.24$
$4.1.25$
$4.1.26$
$4.1.27$
$4.1.28$
$4.1.29$
$4.1.30$
$4.1.31$
$4.1.32$
$4.1.33$ (and $4.1.33$)
$4.1.34$ (and $4.1.34$)
$4.1.35$
$4.1.36$
$4.1.37$
$4.1.38$
$4.1.39$
$4.1.40$
----------
$4.1.46$
$4.1.47$
$4.1.48$
$4.1.49$
$4.1.50$
4.2. Exponential Function
$4.2.1$
$4.2.2$
$4.2.3$
$4.2.4$
$4.2.5$
----------
$4.2.29$
$4.2.30$
$4.2.31$
$4.2.32$
$4.2.33$
$4.2.34$
$4.2.35$
$4.2.36$
$4.2.37$
$4.2.38$
$4.2.39$
4.3. Circular Functions
$4.3.1$
$4.3.2$
$4.3.3$
$4.3.4$
$4.3.5$
$4.3.6$
- - - - -
$4.3.105$
$4.3.106$
$4.3.107$
$4.3.108$
$4.3.109$
$4.3.110$
- - - - -
$4.3.147$ (or $4.3.147$ or $4.3.147$ or $4.3.147$)
4.4. Inverse Circular Functions
4.5. Hyperbolic Functions
$4.5.1$
$4.5.2$
$4.5.3$
$4.5.4$
$4.5.5$
$4.5.6$
$4.5.7$
$4.5.8$
$4.5.9$
$4.5.10$
$4.5.11$
$4.5.12$
$4.5.13$
$4.5.14$
$4.5.15$
$4.5.16$
$4.5.17$
$4.5.18$
$4.5.19$
$4.5.20$
$4.5.21$
$4.5.22$
$4.5.23$
$4.5.24$
$4.5.25$
$4.5.26$
$4.5.27$
$4.5.28$
$4.5.29$
$4.5.30$ (and $4.5.30$ and $4.5.30$
$4.5.31$ (and $4.5.31$
$4.5.32$ (and $4.5.32$ and [[Doubling identity for cosh (3)|$4.5.32$)
4.6. Inverse Hyperbolic Functions
4.7. Use and Extension of the Tables
5. Exponential Integral and Related Functions
5.1. Exponential Integral
$5.1.1$
$5.1.2$
$5.1.3$
$5.1.4$
----------
$5.1.13$
$5.1.14$
5.2. Sine and Cosine Integrals
$5.2.1$
5.3. Use and Extension of the Tables
6. Gamma Function and Related Functions
6.1. Gamma Function
$6.1.1$
$6.1.2$
---------------
6.2. Beta Function
$6.2.1$ (and $6.2.1$ and $6.2.1$)
$6.2.2$ (and $6.2.2$)
6.3. Psi (Digamma Function)
$6.3.1$
$6.3.2$ (and $6.3.2$)
$6.3.3$
$6.3.4$
$6.3.5$
6.4. Polygamma Functions
$6.4.1$ (and $6.4.1$)
$6.4.2$
$6.4.3$
$6.4.4$
$6.4.5$
$6.4.6$
$6.4.7$
$6.4.8$
$6.4.9$
6.5. 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
$7.1.1$
$7.1.2$
$7.1.3$
-------------
$7.1.5$
----------
$7.1.9$
$7.1.10$
-----------
$7.1.14$
$7.1.15$
$7.1.16$
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
-----------
$9.1.2$
$9.1.3$ (and $9.1.3$)
$9.1.4$ (and $9.1.4$)
$9.1.5$ (and $9.1.5$)
-----------
$9.1.10$
-----------
$9.1.64$
$9.1.65$
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
$11.1.1$
$11.1.2$
$11.1.3$
$11.1.4$
$11.1.5$
$11.1.6$
11.2 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 $\mathbf{H}_{\nu}(z)$
--------------------
$12.1.3$
--------------------
$12.1.6$
$12.1.7$
$12.1.8$
$12.1.9$
$12.1.10$
$12.1.11$
$12.1.12$
$12.1.13$
$12.1.14$
$12.1.15$
$12.1.16$
$12.1.17$
12.2 Modified Struve Functions $\mathbf{L}_{\nu}(z)$
12.3 Anger and Weber Functions
$12.3.1$
$12.3.2$
$12.3.3$
$12.3.4$
$12.3.5$
12.4 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
$15.1.1$
$15.1.2$
$15.1.3$
$15.1.4$
$15.1.5$
$15.1.6$
15.2 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
$17.3.1$
----------
$17.3.9$
$17.3.10$
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
23.1 Bernoulli and Euler Polynomials and the Euler-Maclaurin formula
23.2 Riemann Zeta Function and Other Sums of Reciprocal Powers
$23.2.1$
$23.2.2$
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
-------
$27.7.2$
28. Scales of Notation
29. Laplace Transforms
29.1 Definition of the Laplace Transform
$29.1.1$
Subject Index
Index of Notations