Mathematical Introduction

♦ Organization and Objective ♦ Methodology ♦ Notation for the Special Functions ♦ Common Notations and Definitions ♦ Graphics ♦ Applications ♦ Computation ♦ Verification ♦ Special Acknowledgment ♦

Organization and Objective

The mathematical content of the NIST Handbook of Mathematical Functions has been produced over a ten-year period. This part of the project has been carried out by a team comprising the mathematics editor, authors, validators, and the NIST professional staff. Also, valuable initial advice on all aspects of the project was provided by ten external associate editors.

The NIST Handbook has essentially the same objective as the Handbook of Mathematical Functions that was issued in 1964 by the National Bureau of Standards as Number 55 in the NBS Applied Mathematics Series (AMS). This objective is to provide a reference tool for researchers and other users in applied mathematics, the physical sciences, engineering, and elsewhere who encounter special functions in the course of their everyday work.

The mathematical project team has endeavored to take into account the hundreds of research papers and numerous books on special functions that have appeared since 1964. As a consequence, in addition to providing more information about the special functions that were covered in AMS 55, the NIST Handbook includes several special functions that have appeared in the interim in applied mathematics, the physical sciences, and engineering, as well as in other areas. See, for example, Chapters 16, 17, 18, 19, 21, 27, 29, 31, 32, 34, 35, and 36.

Two other ways in which this Handbook differs from AMS 55, and other handbooks, are as follows.

First, the editors instituted a validation process for the whole technical content of each chapter. This process greatly extended normal editorial checking procedures. All chapters went through several drafts (nine in some cases) before the authors, validators, and editors were fully satisfied.

Secondly, as described in the Preface, a Web version (the NIST DLMF) is also available.

Methodology

The first three chapters of the NIST Handbook and DLMF are methodology chapters that provide detailed coverage of, and references for, mathematical topics that are especially important in the theory, computation, and application of special functions. (These chapters can also serve as background material for university graduate courses in complex variables, classical analysis, and numerical analysis.)

Particular care is taken with topics that are not dealt with sufficiently thoroughly from the standpoint of this Handbook in the available literature. These include, for example, multivalued functions of complex variables, for which new definitions of branch points and principal values are supplied (§§1.10(vi), 4.2(i)); the Dirac delta (or delta function), which is introduced in a more readily comprehensible way for mathematicians (§1.17); numerically satisfactory solutions of differential and difference equations (§§2.7(iv), 2.9(i)); and numerical analysis for complex variables (Chapter 3).

In addition, there is a comprehensive account of the great variety of analytical methods that are used for deriving and applying the extremely important asymptotic properties of the special functions, including double asymptotic properties (Chapter 2 and §§10.41(iv), 10.41(v)).

Notation for the Special Functions

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Referenced by:: §1.1, §10.1, §11.1, §12.1, §13.1, §14.1, §15.1, §16.1, §17.1, §18.1(i), §19.1, §20.1, §21.1, §22.1, §23.1, §24.1, §25.1, §26.1, §27.1, §28.1, §29.1, §30.1, §31.1, §32.1, §33.1, §34.1, §35.1, §36.1, §4.1, §5.1, §6.1, §7.1, §8.1, §9.1

The first section in each of the special function chapters (Chapters 5–36) lists notation that has been adopted for the functions in that chapter. This section may also include important alternative notations that have appeared in the literature. With a few exceptions the adopted notations are the same as those in standard applied mathematics and physics literature.

The exceptions are ones for which the existing notations have drawbacks. For example, for the hypergeometric function we often use the notation $\mathbf{F}\left(a,b;c;z\right)$ (§15.2(i)) in place of the more conventional ${{}_{2}F_{1}}\left(a,b;c;z\right)$ or $F\left(a,b;c;z\right)$ . This is because $\mathbf{F}$ is akin to the notation used for Bessel functions (§10.2(ii)), inasmuch as $\mathbf{F}$ is an entire function of each of its parameters $a$ , $b$ , and $c$ : this results in fewer restrictions and simpler equations. Similarly in the case of confluent hypergeometric functions (§13.2(i)).

Other examples are: (a) the notation for the Ferrers functions—also known as associated Legendre functions on the cut—for which existing notations can easily be confused with those for other associated Legendre functions (§14.1); (b) the spherical Bessel functions for which existing notations are unsymmetric and inelegant (§§10.47(i) and 10.47(ii)); and (c) elliptic integrals for which both Legendre’s forms and the more recent symmetric forms are treated fully (Chapter 19).

The Notations section includes all the notations for the special functions adopted in this Handbook. In the corresponding section for the DLMF some of the alternative notations that appear in the first section of the special function chapters are also included.

Common Notations and Definitions

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Defines:: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\left\lceil\NVar{x}\right\rceil$ : ceiling of $x$ , $[\NVar{a},\NVar{b}]$ : closed interval, $[\NVar{a},\NVar{b})$ : half-closed interval, $\mathbb{C}$ : complex plane, $!!$ : double factorial, $\in$ : element of, $\equiv$ : equals by definition, $!$ : factorial (as in $n!$ ), $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $\forall$ : for every, $\inf$ : greatest lower bound (infimum), $\mathbb{Z}$ : set of all integers, $\cap$ : intersection, $\liminf$ : least limit point, $\equiv$ : modular equivalence, $\mod$ : modulo, $\notin$ : not an element of, $\mathbb{N}$ : set of all positive integers, $(\NVar{a},\NVar{b}]$ : half-closed interval, $(\NVar{a},\NVar{b})$ : open interval, $\mathbb{Q}$ : set of all rational numbers, $\mathbb{R}$ : real line, $\setminus$ : set subtraction, $\operatorname{sign}\NVar{x}$ : sign of, $\subset$ : is contained in, $\subseteq$ : is, or is contained in, $\sup$ : least upper bound (supremum) and $\cup$ : union
Referenced by:: Erratum (V1.0.1) for Notation
Amendment (effective with 1.0.1):: The definition of the notation $F(z_{0}e^{2k\pi i})$ was added.

$\mathbb{C}$	complex plane (excluding infinity).
D	decimal places.
$\det$	determinant.
$\delta_{j,k}$ or $\delta_{jk}$	Kronecker delta: 0 if $j\neq k$ ; 1 if $j=k$ .
$\Delta$ (or $\Delta_{x}$ )	forward difference operator: $\Delta f(x)=f(x+1)-f(x)$ .
$\nabla$ (or $\nabla_{x}$ )	backward difference operator: $\nabla f(x)=f(x)-f(x-1)$ . (See also del operator in the Notations section.)
empty sums	zero.
empty products	unity.
$\equiv$	equals by definition.
$\in$	element of.
$\notin$	not an element of.
$\forall$	for every.
$\Longrightarrow$	implies.
$\Longleftrightarrow$	is equivalent to.
$n!$	factorial: $1\cdot 2\cdot 3\cdots n$ if $n=1,2,3,\ldots$ ; 1 if $n=0$ .
$n!!$	double factorial: $2\cdot 4\cdot 6\cdots n$ if $n=2,4,6,\dotsc$ ; $1\cdot 3\cdot 5\cdots n$ if $n=1,3,5,\dotsc$ ; 1 if $n=0,-1$ .
$\left\lfloor x\right\rfloor$	floor or integer part: the integer such that $x-1<\left\lfloor x\right\rfloor\leq x$ , with $x$ real.
$\left\lceil x\right\rceil$	ceiling: the integer such that $x\leq\left\lceil x\right\rceil<x+1$ , with $x$ real.
$F(z_{0}e^{2k\pi i})$	multivalued functions. More generally, $F((z_{0}-a)e^{2k\pi i}+a)$ . See §1.10(vi).
$f(z)\|_{C}=0$	$f(z)$ is continuous at all points of a simple closed contour $C$ in $\mathbb{C}$ .
$<\infty$	is finite, or converges.
$\gg$	much greater than.
$\Im$	imaginary part.
iff	if and only if.
$\inf$	greatest lower bound (infimum).
$\sup$	least upper bound (supremum).
$\cap$	intersection.
$\cup$	union.
$(a,b)$	open interval in $\mathbb{R}$ , or open straight-line segment joining $a$ and $b$ in $\mathbb{C}$ .
$[a,b]$	closed interval in $\mathbb{R}$ , or closed straight-line segment joining $a$ and $b$ in $\mathbb{C}$ .

$(a,b]$ or $[a,b)$	half-closed intervals.
$\subset$	is contained in.
$\subseteq$	is, or is contained in.
$\liminf$	least limit point.
$[a_{j,k}]$ or $[a_{jk}]$	matrix with $(j,k)$ th element $a_{j,k}$ or $a_{jk}$ .
${\mathbf{A}}^{-1}$	inverse of matrix $\mathbf{A}$ .
$\operatorname{tr}\mathbf{A}$	trace of matrix $\mathbf{A}$ .
$\mathbf{A}^{\mathrm{T}}$	transpose of matrix $\mathbf{A}$ .
$\mathbf{I}$	unit matrix.
$\mod$ or modulo	$m\equiv n\pmod{p}$ means $p$ divides $m-n$ , where $m$ , $n$ , and $p$ are positive integers with $m>n$ .
$\mathbb{N}$	set of all positive integers.
${\left(\alpha\right)_{n}}$	Pochhammer’s symbol: $\alpha(\alpha+1)(\alpha+2)\cdots(\alpha+n-1)$ if $n=1,2,3,\dotsc$ ; 1 if $n=0$ .
$\mathbb{Q}$	set of all rational numbers.
$\mathbb{R}$	real line (excluding infinity).
$\Re$	real part.
$\Residue$	residue.
S	significant figures.
$\operatorname{sign}x$	$-1$ if $x<0$ ; 0 if $x=0$ ; 1 if $x>0$ .
$\setminus$	set subtraction.
$\mathbb{Z}$	set of all integers.
$n\mathbb{Z}$	set of all integer multiples of $n$ .

Graphics

Special functions with one real variable are depicted graphically with conventional two-dimensional (2D) line graphs. See, for example, Figures 10.3.1–10.3.4.

With two real variables, special functions are depicted as 3D surfaces, with vertical height corresponding to the value of the function, and coloring added to emphasize the 3D nature. See Figures 10.3.5–10.3.8 for examples.

Special functions with a complex variable are depicted as colored 3D surfaces in a similar way to functions of two real variables, but with the vertical height corresponding to the modulus (absolute value) of the function. See, for example, Figures 5.3.4–5.3.6. However, in many cases the coloring of the surface is chosen instead to indicate the quadrant of the plane to which the phase of the function belongs, thereby achieving a 4D effect. In these cases the phase colors that correspond to the 1st, 2nd, 3rd, and 4th quadrants are arranged in alphabetical order: blue, green, red, and yellow, respectively, and a “Quadrant Colors” icon appears alongside the figure. See, for example, Figures 10.3.9–10.3.16.

Lastly, users may notice some lack of smoothness in the color boundaries of some of the 4D-type surfaces; see, for example, Figure 10.3.9. This nonsmoothness arises because the mesh that was used to generate the figure was optimized only for smoothness of the surface, and not for smoothness of the color boundaries.

Applications

All of the special function chapters include sections devoted to mathematical, physical, and sometimes other applications of the main functions in the chapter. The purpose of these sections is simply to illustrate the importance of the functions in other disciplines; no attempt is made to provide exhaustive coverage.

Computation

All of the special function chapters contain sections that describe available methods for computing the main functions in the chapter, and most also provide references to numerical tables of, and approximations for, these functions. In addition, the DLMF provides references to research papers in which software is developed, together with links to sites where the software can be obtained.

In referring to the numerical tables and approximations we use notation typified by $x=0(.05)1$ , 8D or 8S. This means that the variable $x$ ranges from 0 to 1 in intervals of 0.05, and the corresponding function values are tabulated to 8 decimal places or 8 significant figures.

Another numerical convention is that decimals followed by dots are unrounded; without the dots they are rounded. For example, to 4D $\pi$ is $3.1415\ldots$ (unrounded) and 3.1416 (rounded).

Verification

For all equations and other technical information this Handbook and the DLMF either provide references to the literature for proof or describe steps that can be followed to construct a proof. In the Handbook this information is grouped at the section level and appears under the heading Sources in the References section. In the DLMF this information is provided in pop-up windows at the subsection level.

For equations or other technical information that appeared previously in AMS 55, the DLMF usually includes the corresponding AMS 55 equation number, or other form of reference, together with corrections, if needed. However, none of these citations are to be regarded as supplying proofs.

Special Acknowledgment

I pay tribute to my friend and predecessor Milton Abramowitz. His genius in the creation of the National Bureau of Standards Handbook of Mathematical Functions paid enormous dividends to the world’s scientific, mathematical, and engineering communities, and paved the way for the development of the NIST Handbook of Mathematical Functions and NIST Digital Library of Mathematical Functions.

Frank W. J. Olver, Mathematics Editor