# limits

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## 1—10 of 171 matching pages

##### 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. …
##### 3: 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 $\lim_{z\to 0}\frac{\cosh z-1}{z^{2}}=\frac{1}{2}.$
##### 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: 4.4 Special Values and Limits
###### §4.4(iii) Limits
4.4.14 $\lim_{x\to 0}x^{a}\ln x=0,$ $\Re a>0$,
4.4.18 $\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n}=e.$
##### 6: 18.6 Symmetry, Special Values, and Limits to Monomials
###### §18.6(ii) Limits to Monomials
18.6.2 $\lim_{\alpha\to\infty}\frac{P^{(\alpha,\beta)}_{n}\left(x\right)}{P^{(\alpha,% \beta)}_{n}\left(1\right)}=\left(\frac{1+x}{2}\right)^{n},$
18.6.4 $\lim_{\lambda\to\infty}\frac{C^{(\lambda)}_{n}\left(x\right)}{C^{(\lambda)}_{n% }\left(1\right)}=x^{n},$
18.6.5 $\lim_{\alpha\to\infty}\frac{L^{(\alpha)}_{n}\left(\alpha x\right)}{L^{(\alpha)% }_{n}\left(0\right)}=(1-x)^{n}.$
##### 7: 29.5 Special Cases and Limiting Forms
###### §29.5 Special Cases and Limiting Forms
29.5.4 $\lim_{k\to 1-}a^{m}_{\nu}\left(k^{2}\right)=\lim_{k\to 1-}b^{m+1}_{\nu}\left(k% ^{2}\right)=\nu(\nu+1)-\mu^{2},$
29.5.5 ${\lim_{k\to 1-}\frac{\mathit{Ec}^{m}_{\nu}\left(z,k^{2}\right)}{\mathit{Ec}^{m% }_{\nu}\left(0,k^{2}\right)}=\lim_{k\to 1-}\frac{\mathit{Es}^{m+1}_{\nu}\left(% z,k^{2}\right)}{\mathit{Es}^{m+1}_{\nu}\left(0,k^{2}\right)}}=\frac{1}{(\cosh z% )^{\mu}}F\left({\tfrac{1}{2}\mu-\tfrac{1}{2}\nu,\tfrac{1}{2}\mu+\tfrac{1}{2}% \nu+\tfrac{1}{2}\atop\tfrac{1}{2}};{\tanh}^{2}z\right),$ $m$ even,
$\lim\mathit{Ec}^{m}_{\nu}\left(z,k^{2}\right)=\operatorname{ce}_{m}\left(% \tfrac{1}{2}\pi-z,\theta\right),$
$\lim\mathit{Es}^{m}_{\nu}\left(z,k^{2}\right)=\operatorname{se}_{m}\left(% \tfrac{1}{2}\pi-z,\theta\right),$
##### 8: 18.11 Relations to Other Functions
###### Hermite
The limits (18.11.5)–(18.11.8) hold uniformly for $z$ in any bounded subset of $\mathbb{C}$.
##### 9: 32.14 Combinatorics
32.14.1 $\lim_{N\to\infty}\mathrm{Prob}\left(\frac{\ell_{N}(\boldsymbol{\pi})-2\sqrt{N}% }{N^{1/6}}\leq s\right)=F(s),$
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). …
##### 10: Preface
Thus the utilitarian value of the Handbook will be extended far beyond its original scope and the traditional limitations of printed media. …