# isolated essential

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## 3 matching pages

##### 1: 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*. … ► … ► … ►In any neighborhood of an isolated essential singularity, however small, an analytic function assumes every value in $\u2102$ with at most one exception. …##### 2: 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. …##### 3: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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►Often circumstances allow rather stronger statements, such as uniform convergence, or pointwise convergence at points where $f(x)$ is continuous, with convergence to $(f({x}_{0}-)+f({x}_{0}+))/2$ if ${x}_{0}$ is an isolated point of discontinuity.
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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*. …