limit point and limit circle boundary conditions

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11: 32.14 Combinatorics

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32.14.1 $$\underset{N\to \mathrm{\infty }}{lim}\mathrm{Prob}\left(\frac{{\mathrm{\ell }}_N(𝝅)-2\sqrt{N}}{N^{1/6}}\le s\right)=F(s),$$

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Symbols:

$𝝅(m)$: permutations, ${\mathrm{\ell }}_N(𝝅)$: length and $F(s)$: distribution function

Permalink:

http://dlmf.nist.gov/32.14.E1

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Annotations for §32.14 and Ch.32

… ►The distribution function $F(s)$ given by (32.14.2) arises in random matrix theory where it gives the limiting distribution for the normalized largest eigenvalue in the Gaussian Unitary Ensemble of $n\times n$ Hermitian matrices; see Tracy and Widom (1994). …

12: 3.7 Ordinary Differential Equations

… ►Consideration will be limited to ordinary linear second-order differential equations … ►with limits taken in (3.7.16) when $a$ or $b$, or both, are infinite. … ►

3.7.17 ${lim}_{k\to \mathrm{\infty }}{\lambda }_k=\mathrm{\infty }$.

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Symbols:

${\lambda }_k$: eigenvalues

Permalink:

http://dlmf.nist.gov/3.7.E17

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Annotations for §3.7(iv), §3.7 and Ch.3

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13: 15.11 Riemann’s Differential Equation

… ►Also, if any of $\alpha$, $\beta$, $\gamma$, is at infinity, then we take the corresponding limit in (15.11.1). …

14: 28.7 Analytic Continuation of Eigenvalues

… ►The number of branch points is infinite, but countable, and there are no finite limit points. …

15: 4.4 Special Values and Limits

§4.4 Special Values and Limits

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§4.4(iii) Limits

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4.4.14 $$\underset{x\to 0}{lim}{x}^a\ln x=0,$$ $\mathrm{\Re }a>0$,

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Symbols:

$\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: real part, $a$: real or complex constant and $x$: real variable

A&S Ref:

4.1.31

Permalink:

http://dlmf.nist.gov/4.4.E14

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Annotations for §4.4(iii), §4.4 and Ch.4

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4.4.16 $$\underset{z\to \mathrm{\infty }}{lim}{z}^a{\mathrm{e}}^{-z}=0,$$ $|\mathrm{ph}z|\le \frac{1}{2}\pi -\delta$ (),

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $a$: real or complex constant, $z$: complex variable and $\delta$: constant

A&S Ref:

4.2.20

Permalink:

http://dlmf.nist.gov/4.4.E16

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Annotations for §4.4(iii), §4.4 and Ch.4

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4.4.18 $$\underset{n\to \mathrm{\infty }}{lim}{\left(1+\frac{1}{n}\right)}^n=\mathrm{e}.$$

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Symbols:

$\mathrm{e}$: base of natural logarithm and $n$: integer

A&S Ref:

4.1.17

Permalink:

http://dlmf.nist.gov/4.4.E18

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Annotations for §4.4(iii), §4.4 and Ch.4

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16: Mathematical Introduction

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$(a,b]$ or $[a,b)$	half-closed intervals.
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$lim\; inf$	least limit point.
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17: 4.17 Special Values and Limits

§4.17 Special Values and Limits

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Table 4.17.1: Trigonometric functions: values at multiples of $\frac{1}{12}\pi$.

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$\theta$	$\sin \theta$	$\cos \theta$	$\tan \theta$	$\csc \theta$	$\sec \theta$	$\cot \theta$
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4.17.1 $$\underset{z\to 0}{lim}\frac{\sin z}{z}=1,$$

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Symbols:

$\sin z$: sine function and $z$: complex variable

A&S Ref:

4.3.74

Permalink:

http://dlmf.nist.gov/4.17.E1

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Annotations for §4.17 and Ch.4

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4.17.2 $$\underset{z\to 0}{lim}\frac{\tan z}{z}=1.$$

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Symbols:

$\tan z$: tangent function and $z$: complex variable

A&S Ref:

4.3.75

Permalink:

http://dlmf.nist.gov/4.17.E2

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Annotations for §4.17 and Ch.4

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4.17.3 $$\underset{z\to 0}{lim}\frac{1-\cos z}{z^2}=\frac{1}{2}.$$

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Symbols:

$\cos z$: cosine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.17.E3

Encodings:

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Annotations for §4.17 and Ch.4

18: 28.34 Methods of Computation

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Representations for $w_{\text{I}}(\pi ;a,\pm q)$ with limit formulas for special solutions of the recurrence relations §28.4(ii) for fixed $a$ and $q$; see Schäfke (1961a).

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19: 14.24 Analytic Continuation

… ►the limiting value being taken in (14.24.1) when $2\nu$ is an odd integer. … ►the limiting value being taken in (14.24.4) when $\mu \in \mathbb{Z}$. …

20: 29.5 Special Cases and Limiting Forms

§29.5 Special Cases and Limiting Forms

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29.5.4 $$\underset{k\to 1-}{lim}{a}_{\nu }^m\left(k^2\right)=\underset{k\to 1-}{lim}{b}_{\nu }^{m+1}\left(k^2\right)=\nu (\nu +1)-{\mu }^2,$$

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Symbols:

$a_{\nu }^n\left(k^2\right)$: eigenvalues of Lamé’s equation, $b_{\nu }^n\left(k^2\right)$: eigenvalues of Lamé’s equation, $m$: nonnegative integer, $k$: real parameter, $\nu$: real parameter and $\mu$: parameter

Permalink:

http://dlmf.nist.gov/29.5.E4

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Annotations for §29.5 and Ch.29

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29.5.5 $$\underset{k\to 1-}{lim}\frac{{\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)}{{\mathrm{𝐸𝑐}}_{\nu }^m(0,k^2)}=\underset{k\to 1-}{lim}\frac{{\mathrm{𝐸𝑠}}_{\nu }^{m+1}(z,k^2)}{{\mathrm{𝐸𝑠}}_{\nu }^{m+1}(0,k^2)}=\frac{1}{{(\cosh z)}^{\mu }}F(\genfrac{}{}{0pt}{}{\frac{1}{2}\mu -\frac{1}{2}\nu ,\frac{1}{2}\mu +\frac{1}{2}\nu +\frac{1}{2}}{\frac{1}{2}};{\tanh }^{2}z),$$ $m$ even,

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function, ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$: Lamé function, $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, $m$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\nu$: real parameter and $\mu$: parameter

Permalink:

http://dlmf.nist.gov/29.5.E5

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Annotations for §29.5 and Ch.29

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$lim{\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)={\mathrm{ce}}_m(\frac{1}{2}\pi -z,\theta ),$

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$lim{\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)={\mathrm{se}}_m(\frac{1}{2}\pi -z,\theta ),$

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