# limits to monomials

(0.002 seconds)

## 1—10 of 864 matching pages

##### 1: 18.6 Symmetry, Special Values, and Limits to Monomials
###### §18.6(ii) LimitstoMonomials
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 $\lim_{\alpha\to\infty}\frac{L^{(\alpha)}_{n}\left(\alpha x\right)}{L^{(\alpha)% }_{n}\left(0\right)}=(1-x)^{n}.$
##### 2: 33.18 Limiting Forms for Large $\ell$
###### §33.18 Limiting Forms for Large $\ell$
As $\ell\to\infty$ with $\epsilon$ and $r$ ($\neq 0$) fixed, …
##### 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: 4.4 Special Values and Limits
###### §4.4(iii) Limits
4.4.17 $\lim_{n\to\infty}\left(1+\frac{z}{n}\right)^{n}=e^{z},$ $z=$ constant.
4.4.18 $\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n}=e.$
##### 5: 35.11 Tables
Each table expresses the zonal polynomials as linear combinations of monomial symmetric functions.
##### 6: 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,
29.5.6 $\lim_{k\to 1-}\frac{\mathit{Ec}^{m}_{\nu}\left(z,k^{2}\right)}{\left.\ifrac{% \mathrm{d}\mathit{Ec}^{m}_{\nu}\left(z,k^{2}\right)}{\mathrm{d}z}\right|_{z=0}% }=\lim_{k\to 1-}\frac{\mathit{Es}^{m+1}_{\nu}\left(z,k^{2}\right)}{\left.% \ifrac{\mathrm{d}\mathit{Es}^{m+1}_{\nu}\left(z,k^{2}\right)}{\mathrm{d}z}% \right|_{z=0}}=\frac{\tanh z}{(\cosh z)^{\mu}}F\left({\tfrac{1}{2}\mu-\tfrac{1% }{2}\nu+\tfrac{1}{2},\tfrac{1}{2}\mu+\tfrac{1}{2}\nu+1\atop\tfrac{3}{2}};{% \tanh}^{2}z\right),$ $m$ odd,
If $k\to 0+$ and $\nu\to\infty$ in such a way that $k^{2}\nu(\nu+1)=4\theta$ (a positive constant), then …
##### 7: 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 $\lim_{z\to 0}\frac{1-\cos z}{z^{2}}=\frac{1}{2}.$
##### 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: Preface
Abramowitz and Stegun’s Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables is being completely rewritten with regard to the needs of today. …The DLMF will make full use of advanced communications and computational resources to present downloadable math data, manipulable graphs, tables of numerical values, and math-aware search. … Thus the utilitarian value of the Handbook will be extended far beyond its original scope and the traditional limitations of printed media. …
##### 10: 32.14 Combinatorics
With $1\leq m_{1}<\cdots, $\boldsymbol{\pi}(m_{1}),\boldsymbol{\pi}(m_{2}),\dots,\boldsymbol{\pi}(m_{n})$ is said to be an increasing subsequence of $\boldsymbol{\pi}$ of length $n$ when $\boldsymbol{\pi}(m_{1})<\boldsymbol{\pi}(m_{2})<\cdots<\boldsymbol{\pi}(m_{n})$. …
32.14.1 $\lim_{N\to\infty}\mathrm{Prob}\left(\frac{\ell_{N}(\boldsymbol{\pi})-2\sqrt{N}% }{N^{1/6}}\leq s\right)=F(s),$
32.14.3 $w(x)\sim\operatorname{Ai}\left(x\right),$ $x\to+\infty$,
32.14.4 $w(x)\sim\sqrt{-\tfrac{1}{2}x},$ $x\to-\infty$,
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). …