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1: 37.20 Mathematical Applications

… ►Partial sums of Fourier orthogonal polynomial expansions are polynomials of best approximation in $L^2\left(W\right)$ space and they are also the essential building blocks for approximation in $L^p$ spaces. …OPs are essential for developing approximation theory on regular domains, including characterization of best approximation. … ►Fourier orthogonal expansions provide essential tools and building blocks for harmonic analysis and computational harmonic analysis. …

2: 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. …

3: 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. …

4: 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. …

5: 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. …

6: 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: …

7: Preface

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

8: 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. …

9: 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. …

10: 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 $x\to -1-$. …