2 Existing Scope and Content

The 29 chapters of the existing handbook are listed in Table 1. Except for a few chapters devoted to introduction, methodology or application, a typical chapter addresses a particular class of special functions.

Table 1: Chapters of the NBS Handbook of Mathematical Functions.

Chapter	Title
1	Mathematical Constants
2	Physical Constants and Conversion Factors
3	Elementary Analytical Methods
4	Elementary Transcendental Functions
5	Exponential Integral and Related Functions
6	Gamma Function and Related Functions
7	Error Function and Fresnel Integrals
8	Legendre Functions
9	Bessel Functions of Integer Order
10	Bessel Functions of Fractional Order
11	Integrals of Bessel Functions
12	Struve Functions and Related Functions
13	Confluent Hypergeometric Functions
14	Coulomb Wave Functions
15	Hypergeometric Functions
16	Jacobian Elliptic Functions and Theta Functions
17	Elliptic Integrals
18	Weierstrass Elliptic and Related Functions
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
25	Numerical Interpolation, Differentiation and Integration
26	Probability Functions
27	Miscellaneous Functions
28	Scales of Notation
29	Laplace Transforms

Five chapters are introductory. Chapters 1 and 2 provide mathematical and physical constants, and conversion factors between metric and US customary units. Chapter 3 covers binomials, progressions, and means; inequalities; formulas from calculus and complex analysis; absolute and relative errors; infinite series; solution of quadratic, cubic and quartic equations; successive approximation methods; and continued fractions. Chapter 4 covers the logarithmic, exponential, circular, inverse circular, hyperbolic and inverse hyperbolic functions. Chapter 28 give basic information on computer arithmetic.

Numerical analysis is the subject of the methodological Chapter 25. It covers forward, central, mean, divided and reciprocal differences; Lagrange, Newton, trigonometric and inverse interpolation, including formulas with throwback and stencils for bivariate interpolation; differentiation formulas, including stencils for partial derivatives and the Laplacian and biharmonic operators; integration formulas, including open and closed Newton-Cotes, Gaussian, and other types of quadrature, and numerous stencils for multidimensional integration; and numerical solution of ordinary differential equations, including Runge-Kutta and predictor-corrector methods. For nearly all formulas in Chapter 25, the corresponding error order, error estimate or error bound is given.

Three chapters are oriented toward applications. Chapter 24 addresses combinatorial analysis (binomial and multinomial coefficients; partitions of integers; and Mobius and Euler functions). Chapter 26 deals with statistics (definitions and properties of distribution functions; normal and bivariate normal probability functions; and chi-square, incomplete beta, variance-ratio, and Student's $t$-distributions). Finally, chapter 29 concerns Laplace transforms (definitions and formulas; and short tables of Laplace and Laplace-Stieltjes transforms).

The remaining 20 chapters form the core material of the handbook. Each treats an individual class of special functions, and typically each is divided into four sections. The first section, Mathematical Properties, presents definitions, differential equations, integral representations, recurrence relations, functional relations, series expansions, continued fractions, asymptotic expansions, polynomial approximations, special values, derivative formulas, integrals, zeros, graphical representations, and so on. The second section, Numerical Methods, gives advice on how to compute function values effectively by using numerical tables, interpolation methods, and relevant mathematical properties such as recurrence relations. The third section, References, is divided into two parts, one for texts and the other for tables. The final section, Tables, contains data taken from previously published sources, verified and augmented with computations performed at NBS.