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1: 9.14 Incomplete Airy Functions

… ►Incomplete Airy functions are defined by the contour integral (9.5.4) when one of the integration limits is replaced by a variable real or complex parameter. …

2: 33.18 Limiting Forms for Large $\mathrm{\ell }$

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

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3: 4.31 Special Values and Limits

§4.31 Special Values and Limits

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Table 4.31.1: Hyperbolic functions: values at multiples of $\frac{1}{2}\pi \mathrm{i}$.

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$z$	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 $$\underset{z\to 0}{lim}\frac{\sinh z}{z}=1,$$

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Symbols:

$\sinh z$: hyperbolic sine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.31.E1

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Annotations for §4.31 and Ch.4

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

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Symbols:

$\tanh z$: hyperbolic tangent function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.31.E2

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Annotations for §4.31 and Ch.4

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

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Symbols:

$\cosh z$: hyperbolic cosine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.31.E3

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Annotations for §4.31 and Ch.4

4: 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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4.4.15 $$\underset{x\to \mathrm{\infty }}{lim}{x}^a{\mathrm{e}}^{-x}=0,$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $a$: real or complex constant and $x$: real variable

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http://dlmf.nist.gov/4.4.E15

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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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5: 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

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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 $$\underset{k\to 1-}{lim}\frac{{\mathit{Ec}}_{\nu }^m(z,k^2)}{{\mathit{Ec}}_{\nu }^m(0,k^2)}=\underset{k\to 1-}{lim}\frac{{\mathit{Es}}_{\nu }^{m+1}(z,k^2)}{{\mathit{Es}}_{\nu }^{m+1}(0,k^2)}=\frac{1}{{(\cosh z)}^{\mu }}F(\genfrac{}{}{0pt}{}{\frac{1}{2}\mu -\frac{1}{2}\nu ,\frac{1}{2}\mu +\frac{1}{2}\nu +\frac{1}{2}}{\frac{1}{2}};{\tanh }^{2}z),$$ $m$ 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, ${\mathit{Ec}}_{\nu }^m(z,k^2)$: Lamé function, ${\mathit{Es}}_{\nu }^m(z,k^2)$: 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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$lim{\mathit{Ec}}_{\nu }^m(z,k^2)={\mathrm{ce}}_m(\frac{1}{2}\pi -z,\theta ),$

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

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6: 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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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

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7: 32.14 Combinatorics

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

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Symbols:

$\mathit{\pi }(m)$: permutations, ${\mathrm{\ell }}_N(\mathit{\pi })$: 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). …

8: Preface

… ►Thus the utilitarian value of the Handbook will be extended far beyond its original scope and the traditional limitations of printed media. …

9: 26.5 Lattice Paths: Catalan Numbers

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§26.5(iv) Limiting Forms

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26.5.7 $$\underset{n\to \mathrm{\infty }}{lim}\frac{C\left(n+1\right)}{C\left(n\right)}=4.$$

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Symbols:

$C\left(n\right)$: Catalan number and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.5.E7

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Annotations for §26.5(iv), §26.5 and Ch.26

10: 3.9 Acceleration of Convergence

… ►A transformation of a convergent sequence $\{s_n\}$ with limit $\sigma$ into a sequence $\{t_n\}$ is called limit-preserving if $\{t_n\}$ converges to the same limit $\sigma$. ►The transformation is accelerating if it is limit-preserving and if …Similarly for convergent series if we regard the sum as the limit of the sequence of partial sums. … ►It may even fail altogether by not being limit-preserving. …