# limit point and limit circle boundary conditions

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##### 1: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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►By

*Weyl’s alternative*${n}_{1}$ equals either 1 (the*limit point case*) or 2 (the*limit circle case*), and similarly for ${n}_{2}$. … A*boundary value*for the end point $a$ is a linear form $\mathcal{B}$ on $\mathcal{D}({\mathcal{L}}^{\ast})$ of the form … ►The above results, especially the discussions of deficiency indices and limit point and limit circle boundary conditions, lay the basis for further applications. … ►The materials developed here follow from the extensions of the Sturm–Liouville theory of second order ODEs as developed by Weyl, to include the*limit point and limit circle*singular cases. …See, in particular, the overview Everitt (2005b, pp. 45–74), and the uniformly annotated listing of $51$ solved Sturm–Liouville problems in Everitt (2005a, pp. 272–331), each with their limit point, or circle, boundary behaviors categorized.##### 2: 1.9 Calculus of a Complex Variable

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###### Continuity

… ►A point ${z}_{0}$ is a*limit point*(*limiting point*or*accumulation point*) of a set of points $S$ in $\u2102$ (or $\u2102\cup \mathrm{\infty}$) if every neighborhood of ${z}_{0}$ contains a point of $S$ distinct from ${z}_{0}$. …As a consequence, every neighborhood of a limit point of $S$ contains an infinite number of points of $S$. Also, the union of $S$ and its limit points is the*closure*of $S$. … ►A function $f(z)$ is*complex differentiable*at a point $z$ if the following limit exists: …##### 3: 20.13 Physical Applications

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►In the singular limit
$\mathrm{\Im}\tau \to 0+$, the functions ${\theta}_{j}\left(z\right|\tau )$, $j=1,2,3,4$, become integral kernels of Feynman path integrals (distribution-valued Green’s functions); see Schulman (1981, pp. 194–195).
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##### 4: 2.1 Definitions and Elementary Properties

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►Let $\mathbf{X}$ be a point set with a limit point
$c$.
As $x\to c$ in $\mathbf{X}$
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►If $c$ is a finite limit point of $\mathbf{X}$, then
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►Similarly for finite limit point
$c$ in place of $\mathrm{\infty}$.
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►where $c$ is a finite, or infinite, limit point of $\mathbf{X}$.
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##### 5: 33.18 Limiting Forms for Large $\mathrm{\ell}$

###### §33.18 Limiting Forms for Large $\mathrm{\ell}$

…##### 6: 1.4 Calculus of One Variable

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►When this limit exists $f$ is

*differentiable*at $x$. … ►when the last limit exists. … ►If the limit exists then $f$ is called*Riemann integrable*. … ►when this limit exists. … ►when this limit exists. …##### 7: 4.31 Special Values and Limits

###### §4.31 Special Values and Limits

►$z$ | 0 | $\frac{1}{2}\pi \mathrm{i}$ | $\pi \mathrm{i}$ | $\frac{3}{2}\pi \mathrm{i}$ | $\mathrm{\infty}$ |
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4.31.1
$$\underset{z\to 0}{lim}\frac{\mathrm{sinh}z}{z}=1,$$

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4.31.2
$$\underset{z\to 0}{lim}\frac{\mathrm{tanh}z}{z}=1,$$

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4.31.3
$$\underset{z\to 0}{lim}\frac{\mathrm{cosh}z-1}{{z}^{2}}=\frac{1}{2}.$$

##### 8: 10.72 Mathematical Applications

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►In regions in which (10.72.1) has a simple turning point
${z}_{0}$, that is, $f(z)$ and $g(z)$ are analytic (or with weaker conditions if $z=x$ is a real variable) and ${z}_{0}$ is a simple zero of $f(z)$, asymptotic expansions of the solutions $w$ for large $u$ can be constructed in terms of Airy functions or equivalently Bessel functions or modified Bessel functions of order $\frac{1}{3}$ (§9.6(i)).
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►In regions in which the function $f(z)$ has a simple pole at $z={z}_{0}$ and ${(z-{z}_{0})}^{2}g(z)$ is analytic at $z={z}_{0}$ (the case $\lambda =-1$ in §10.72(i)), asymptotic expansions of the solutions $w$ of (10.72.1) for large $u$ can be constructed in terms of Bessel functions and modified Bessel functions of order $\pm \sqrt{1+4\rho}$, where $\rho $ is the limiting value of ${(z-{z}_{0})}^{2}g(z)$ as $z\to {z}_{0}$.
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##### 9: 2.4 Contour Integrals

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2.4.3
$$q(t)=\frac{1}{2\pi \mathrm{i}}\underset{\eta \to \mathrm{\infty}}{lim}{\int}_{\sigma -\mathrm{i}\eta}^{\sigma +\mathrm{i}\eta}{\mathrm{e}}^{tz}Q(z)dz,$$
$$,

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►is seen to converge absolutely at each limit, and be independent of $\sigma \in [c,\mathrm{\infty})$.
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►If this integral converges uniformly at each limit for all sufficiently large $t$, then by the Riemann–Lebesgue lemma (§1.8(i))
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►in which $a$ is finite, $b$ is finite or infinite, and $\omega $ is the angle of slope of $\mathcal{P}$ at $a$, that is, $lim(\mathrm{ph}\left(t-a\right))$ as $t\to a$ along $\mathcal{P}$.
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►The branch of ${\omega}_{0}=\mathrm{ph}\left({p}^{\prime \prime}({t}_{0})\right)$ is the one satisfying $|\theta +2\omega +{\omega}_{0}|\le \frac{1}{2}\pi $, where $\omega $ is the limiting value of $\mathrm{ph}\left(t-{t}_{0}\right)$ as $t\to {t}_{0}$ from $b$.
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##### 10: 1.10 Functions of a Complex Variable

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►If ${f}_{2}(z)$, analytic in ${D}_{2}$, equals ${f}_{1}(z)$ on an arc in $D={D}_{1}\cap {D}_{2}$, or on just an infinite number of points with a limit point in $D$, then they are equal throughout $D$ and ${f}_{2}(z)$ is called an

*analytic continuation*of ${f}_{1}(z)$. … ►An isolated singularity ${z}_{0}$ is always removable when ${lim}_{z\to {z}_{0}}f(z)$ exists, for example $(\mathrm{sin}z)/z$ at $z=0$. … ►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. … ►If the path circles a branch point at $z=a$, $k$ times in the positive sense, and returns to ${z}_{0}$ without encircling any other branch point, then its value is denoted conventionally as $F(({z}_{0}-a){\mathrm{e}}^{2k\pi \mathrm{i}}+a)$. …