# essential

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## 1—10 of 20 matching pages

##### 1: 14.32 Methods of Computation

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►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.
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##### 2: 34.12 Physical Applications

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►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.
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##### 3: 1.10 Functions of a Complex Variable

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►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 $\u2102$ with at most one exception. …##### 4: Notices

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►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.
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##### 5: Guide to Searching the DLMF

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

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► Stegun, editors); and to disseminate essentially the same information from a public website operated by NIST.
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##### 7: 32.2 Differential Equations

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►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

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►An essential feature of such symmetric operators is that their eigenvalues $\lambda $ are real, and eigenfunctions
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►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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►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 $x\to -1-$.
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##### 10: Mathematical Introduction

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►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). …