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1: 14.32 Methods of Computation

… ►Essentially the same comments that are made in §15.19 concerning the computation of hypergeometric functions apply to the functions described in the present chapter. …

2: 34.12 Physical Applications

… ►The angular momentum coupling coefficients ($\mathit{3}j$, $\mathit{6}j$, and $\mathit{9}j$ symbols) are essential in the fields of nuclear, atomic, and molecular physics. …

3: 1.10 Functions of a Complex Variable

… ►Lastly, if $a_n\ne 0$ for infinitely many negative $n$, then $z_0$ is an isolated essential singularity. … ► … ►If the poles are infinite in number, then the point at infinity is called an essential singularity: it is the limit point of the poles. … ►In any neighborhood of an isolated essential singularity, however small, an analytic function assumes every value in $ℂ$ with at most one exception. …

4: Notices

… ►The DLMF wishes to provide users of special functions with essential reference information related to the use and application of special functions in research, development, and education. …

5: Guide to Searching the DLMF

… ►DLMF search recognizes just the essential font differences, that is, the font style differences deemed important for the DLMF contents: …

6: Preface

… ► Stegun, editors); and to disseminate essentially the same information from a public website operated by NIST. …

7: 32.2 Differential Equations

… ►An equation is said to have the Painlevé property if all its solutions are free from movable branch points; the solutions may have movable poles or movable isolated essential singularities (§1.10(iii)), however. …

8: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►An essential feature of such symmetric operators is that their eigenvalues $\lambda$ are real, and eigenfunctions … ►In unusual cases $N=\mathrm{\infty }$, even for all $\mathrm{\ell }$, such as in the case of the Schrödinger–Coulomb problem ($V=-r^{-1}$) discussed in §18.39 and §33.14, where the point spectrum actually accumulates at the onset of the continuum at $\lambda =0$, implying an essential singularity, as well as a branch point, in matrix elements of the resolvent, (1.18.66). … ►If $T\subset T^{\ast \ast }=T^{\ast }$ then $T$ is essentially self-adjoint and if $T=T^{\ast }$ then $T$ is self-adjoint. …

9: 18.39 Applications in the Physical Sciences

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Figure 18.39.2: Coulomb–Pollaczek weight functions, $x\in [-1,1]$, (18.39.50) for $s=10$, $l=0$, and $Z=\pm 1$. For $Z=+1$ the weight function, red curve, has an essential singularity at $x=-1$, as all derivatives vanish as $x\to -1+$; the green curve is ${\int }_{-1}^x{w}^{\mathrm{CP}}(y)dy$, to be compared with its histogram approximation in §18.40(ii). For $Z=-1$ the weight function, blue curve, is non-zero at $x=-1$, but this point is also an essential singularity as the discrete parts of the weight function of (18.39.51) accumulate as $k\to \mathrm{\infty }$, $x_k\to -1-$. Magnify

… ►The Schrödinger operator essential singularity, seen in the accumulation of discrete eigenvalues for the attractive Coulomb problem, is mirrored in the accumulation of jumps in the discrete Pollaczek–Stieltjes measure as $x\to -1-$. …

10: Mathematical Introduction

… ►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). …