limits to monomials

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1: 18.6 Symmetry, Special Values, and Limits to Monomials

§18.6 Symmetry, Special Values, and Limits to Monomials

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§18.6(ii) Limits to Monomials

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18.6.2 $$\underset{\alpha \to \mathrm{\infty }}{lim}\frac{P_n^{(\alpha ,\beta )}\left(x\right)}{P_n^{(\alpha ,\beta )}\left(1\right)}={\left(\frac{1+x}{2}\right)}^n,$$

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Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.6(ii)

Permalink:

http://dlmf.nist.gov/18.6.E2

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Annotations for §18.6(ii), §18.6 and Ch.18

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18.6.4 $$\underset{\lambda \to \mathrm{\infty }}{lim}\frac{C_n^{(\lambda )}\left(x\right)}{C_n^{(\lambda )}\left(1\right)}=x^n,$$

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Symbols:

$C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.18(iv), §18.6(ii)

Permalink:

http://dlmf.nist.gov/18.6.E4

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Annotations for §18.6(ii), §18.6 and Ch.18

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18.6.5 $$\underset{\alpha \to \mathrm{\infty }}{lim}\frac{L_n^{(\alpha )}\left(\alpha x\right)}{L_n^{(\alpha )}\left(0\right)}={(1-x)}^n.$$

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.6(ii)

Permalink:

http://dlmf.nist.gov/18.6.E5

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Annotations for §18.6(ii), §18.6 and Ch.18

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

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

►As $\mathrm{\ell }\to \mathrm{\infty }$ with $ϵ$ and $r$ ($\ne 0$) fixed, …

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

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Symbols:

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

Permalink:

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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4.4.17 $$\underset{n\to \mathrm{\infty }}{lim}{\left(1+\frac{z}{n}\right)}^n={\mathrm{e}}^z,$$ $z=$ constant.

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Symbols:

$\mathrm{e}$: base of natural logarithm, $n$: integer and $z$: complex variable

A&S Ref:

4.2.21

Permalink:

http://dlmf.nist.gov/4.4.E17

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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: 37.20 Mathematical Applications

… ►The $L^2$ norms of the monic OPs are the error of the least square approximation of monomials by polynomials of lower degrees. … ►For regular domains, such as square, sphere, ball, simplex, and conic domains, they are used to study convolution structure, maximal functions, and interpolation spaces, as well as localized kernel and localized frames. … ►Hermite polynomials on ${ℝ}^d$ (see §37.17) are closely related to the $d$-dimensional harmonic oscillator, see for instance Aquilanti et al. (1997). … ►The nodes of these cubature rules are closely related to common zeros of OPs and they are often good points for polynomial interpolation. …

6: 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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29.5.6 $$\underset{k\to 1-}{lim}\frac{{\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)}{{d{\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)/dz|}_{z=0}}=\underset{k\to 1-}{lim}\frac{{\mathrm{𝐸𝑠}}_{\nu }^{m+1}(z,k^2)}{{d{\mathrm{𝐸𝑠}}_{\nu }^{m+1}(z,k^2)/dz|}_{z=0}}=\frac{\tanh z}{{(\cosh z)}^{\mu }}F(\genfrac{}{}{0pt}{}{\frac{1}{2}\mu -\frac{1}{2}\nu +\frac{1}{2},\frac{1}{2}\mu +\frac{1}{2}\nu +1}{\frac{3}{2}};{\tanh }^{2}z),$$ $m$ odd,

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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, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\cosh z$: 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.E6

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Annotations for §29.5 and Ch.29

… ►If $k\to 0+$ and $\nu \to \mathrm{\infty }$ 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

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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

8: 18.11 Relations to Other Functions

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§18.11(ii) Formulas of Mehler–Heine Type

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Jacobi

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Laguerre

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Hermite

… ►The limits (18.11.5)–(18.11.8) hold uniformly for $z$ in any bounded subset of $ℂ$.

9: 35.11 Tables

… ►Each table expresses the zonal polynomials as linear combinations of monomial symmetric functions.

10: 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. …