# limit circle

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##### 1: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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 $\mathcal{D}({\mathcal{L}}^{*})$ of the form
1.18.71 $\mathcal{B}(f)=\lim_{x\to a+}\left(\alpha(x)f(x)+\beta(x)f^{\prime}(x)\right),$ $f\in\mathcal{D}({\mathcal{L}}^{*})$,
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. …
##### 2: 4.31 Special Values and Limits
###### §4.31 Special Values and Limits
4.31.1 $\lim_{z\to 0}\frac{\sinh z}{z}=1,$
4.31.2 $\lim_{z\to 0}\frac{\tanh z}{z}=1,$
4.31.3 $\lim_{z\to 0}\frac{\cosh z-1}{z^{2}}=\frac{1}{2}.$
##### 3: 26.5 Lattice Paths: Catalan Numbers
###### §26.5(iv) Limiting Forms
26.5.7 $\lim_{n\to\infty}\frac{C\left(n+1\right)}{C\left(n\right)}=4.$
##### 4: 4.17 Special Values and Limits
###### §4.17 Special Values and Limits
4.17.1 $\lim_{z\to 0}\frac{\sin z}{z}=1,$
4.17.2 $\lim_{z\to 0}\frac{\tan z}{z}=1.$
4.17.3 $\lim_{z\to 0}\frac{1-\cos z}{z^{2}}=\frac{1}{2}.$
##### 5: 35.2 Laplace Transform
Assume that $\int_{\boldsymbol{\mathcal{S}}}\left|g(\mathbf{U}+\mathrm{i}\mathbf{V})\right|% \,\mathrm{d}{\mathbf{V}}$ converges, and also that its limit as $\mathbf{U}\to\infty$ is $0$. …
35.2.2 $f(\mathbf{X})=\dfrac{1}{(2\pi\mathrm{i})^{m(m+1)/2}}\int\operatorname{etr}% \left(\mathbf{Z}\mathbf{X}\right)g(\mathbf{Z})\,\mathrm{d}{\mathbf{Z}},$
##### 6: 10.34 Analytic Continuation
10.34.1 $I_{\nu}\left(ze^{m\pi i}\right)=e^{m\nu\pi i}I_{\nu}\left(z\right),$
10.34.2 $K_{\nu}\left(ze^{m\pi i}\right)=e^{-m\nu\pi i}K_{\nu}\left(z\right)-\pi i\sin% \left(m\nu\pi\right)\csc\left(\nu\pi\right)I_{\nu}\left(z\right).$
10.34.4 $K_{\nu}\left(ze^{m\pi i}\right)=\csc\left(\nu\pi\right)\left(\pm\sin\left(m\nu% \pi\right)K_{\nu}\left(ze^{\pm\pi i}\right)\mp\sin\left((m\mp 1)\nu\pi\right)K% _{\nu}\left(z\right)\right).$
If $\nu=n(\in\mathbb{Z})$, then limiting values are taken in (10.34.2) and (10.34.4):
10.34.5 $K_{n}\left(ze^{m\pi i}\right)=(-1)^{mn}K_{n}\left(z\right)+(-1)^{n(m-1)-1}m\pi iI% _{n}\left(z\right),$
##### 7: 26.10 Integer Partitions: Other Restrictions
###### §26.10(v) Limiting Form
26.10.16 $p\left(\mathcal{D},n\right)\sim\frac{{\mathrm{e}}^{\pi\sqrt{n/3}}}{(768n^{3})^% {1/4}},$ $n\to\infty$.
26.10.17 $p\left(\mathcal{D},n\right)=\pi\sum_{k=1}^{\infty}\frac{A_{2k-1}(n)}{(2k-1)% \sqrt{24n+1}}I_{1}\left(\frac{\pi}{2k-1}\sqrt{\frac{24n+1}{72}}\right),$
26.10.18 $A_{k}(n)=\sum_{\begin{subarray}{c}1
##### 8: 19.20 Special Cases
$\lim_{p\to 0+}\sqrt{p}R_{J}\left(0,y,z,p\right)=\frac{3\pi}{2\sqrt{y}\sqrt{z}},$
$\lim_{p\to 0-}R_{J}\left(0,y,z,p\right)={-R_{D}\left(0,y,z\right)-R_{D}\left(0% ,z,y\right)}=\frac{-6}{yz}R_{G}\left(0,y,z\right).$
##### 9: 2.10 Sums and Sequences
(5.11.7) shows that the integrals around the large quarter circles vanish as $n\to\infty$. …
• (b´)

On the circle $|z|=r$, the function $f(z)-g(z)$ has a finite number of singularities, and at each singularity $z_{j}$, say,

2.10.30 $f(z)-g(z)=O\left((z-z_{j})^{\sigma_{j}-1}\right),$ $z\to z_{j}$,

where $\sigma_{j}$ is a positive constant.

• The singularities of $f(z)$ on the unit circle are branch points at $z=e^{\pm i\alpha}$. To match the limiting behavior of $f(z)$ at these points we set … For uniform expansions when two singularities coalesce on the circle of convergence see Wong and Zhao (2005). …