# isolated essential

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## 3 matching pages

##### 1: 1.10 Functions of a Complex Variable
Lastly, if $a_{n}\not=0$ 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 $\mathbb{C}$ with at most one exception. …
##### 2: 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. …
##### 3: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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. … An essential feature of such symmetric operators is that their eigenvalues $\lambda$ are real, and eigenfunctions … In unusual cases $N=\infty$, even for all $\ell$, 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 $T\subset T^{**}={T}^{*}$ then $T$ is essentially self-adjoint and if $T={T}^{*}$ then $T$ is self-adjoint. …