limiting form

SearchAdvancedHelp

(0.002 seconds)

31—40 of 78 matching pages

31: 11.9 Lommel Functions

… ►the right-hand side being replaced by its limiting form when $\mu \pm \nu$ is an odd negative integer. …

32: 16.4 Argument Unity

… ►with limiting form $a\left(\psi \left(a+n+1\right)-\psi \left(a\right)\right)=\frac{a\frac{d}{da}{\left(a\right)}_{n+1}}{{\left(a\right)}_{n+1}}$ in the case that $c=a+1$. …

33: 8.19 Generalized Exponential Integral

… ►The right-hand sides are replaced by their limiting forms when $p=1,2,3,\mathrm{\dots }$. …

34: 13.23 Integrals

…

35: 3.3 Interpolation

… ►For example, for $k+1$ coincident points the limiting form is given by $\left[z_0,z_0,\mathrm{\dots },z_0\right]f=f^{(k)}(z_0)/k!$. …

36: 1.9 Calculus of a Complex Variable

… ►or its limiting form, and is invariant under bilinear transformations. …

37: Errata

… ►

Originally the limiting form for $\mathrm{sc}(z,k)$ in the last line of this table was incorrect ($\cosh z$, instead of $\sinh z$).

$\mathrm{sn}(z,k)$ $\to$	$\tanh z$	$\mathrm{cd}(z,k)$ $\to$	$1$	$\mathrm{dc}(z,k)$ $\to$	$1$	$\mathrm{ns}(z,k)$ $\to$	$\coth z$
$\mathrm{cn}(z,k)$ $\to$	$\mathrm{sech}z$	$\mathrm{sd}(z,k)$ $\to$	$\sinh z$	$\mathrm{nc}(z,k)$ $\to$	$\cosh z$	$\mathrm{ds}(z,k)$ $\to$	$\mathrm{csch}z$
$\mathrm{dn}(z,k)$ $\to$	$\mathrm{sech}z$	$\mathrm{nd}(z,k)$ $\to$	$\cosh z$	$\mathrm{sc}(z,k)$ $\to$	$\sinh z$	$\mathrm{cs}(z,k)$ $\to$	$\mathrm{csch}z$

Reported 2010-11-23.

…

38: 18.28 Askey–Wilson Class

… ►

18.28.29 $$\underset{q\to 1}{lim}\frac{p_n(1-\frac{1}{2}x{(1-q)}^2;q^a,q^b,q^c,q^d|q)}{{(1-q)}^{3n}}=W_n(x;a,b,c,d).$$

ⓘ

Symbols:

$p_n(x;a,b,c,d|q)$: Askey–Wilson polynomial, $W_n(x;a,b,c,d)$: Wilson polynomial, $q$: real variable, $n$: nonnegative integer and $x$: real variable

Proof sketch:

Substitute (18.28.1) in the left-hand side, take the limit, and use (18.26.1).

Notes:

Another form of this limit is given in Koekoek et al. (2010, (14.1.21)).

Referenced by:

§18.28(x)

Permalink:

http://dlmf.nist.gov/18.28.E29

Encodings:

pMML, png, TeX

See also:

Annotations for §18.28(x), §18.28(x), §18.28 and Ch.18

…

39: 19.14 Reduction of General Elliptic Integrals

… ►The choice among 21 transformations for final reduction to Legendre’s normal form depends on inequalities involving the limits of integration and the zeros of the cubic or quartic polynomial. …

40: 28.28 Integrals, Integral Representations, and Integral Equations

… ►

28.28.21 $$\frac{4}{\pi }{\int }_0^{\pi /2}{𝒞}_{2\mathrm{\ell }+1}^{(j)}(2hR)\cos \left((2\mathrm{\ell }+1)\varphi \right){\mathrm{ce}}_{2m+1}(t,h^2)dt={(-1)}^{\mathrm{\ell }+m}{A}_{2\mathrm{\ell }+1}^{2m+1}(h^2){\mathrm{Mc}}_{2m+1}^{(j)}(z,h),$$

ⓘ

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions, $\varphi (z,t)$: function, $R(z,t)$: function and $A_m(q)$: Fourier coefficient

A&S Ref:

20.7.26 (in different form and only for $\mathrm{\ell }=0$)

Referenced by:

Erratum (V1.0.11) for Equations (28.28.21) and (28.28.22)

Permalink:

http://dlmf.nist.gov/28.28.E21

Encodings:

pMML, png, TeX

Errata (effective with 1.0.11):

Originally the prefactor and uppper limit of integration were given incorrectly as ${\displaystyle \frac{2}{\pi }}{\int }_0^{\pi }$.

Reported 2015-05-20 by Ruslan Kabasayev

See also:

Annotations for §28.28(ii), §28.28 and Ch.28

►

28.28.22 $$\frac{4}{\pi }{\int }_0^{\pi /2}{𝒞}_{2\mathrm{\ell }+1}^{(j)}(2hR)\sin \left((2\mathrm{\ell }+1)\varphi \right){\mathrm{se}}_{2m+1}(t,h^2)dt={(-1)}^{\mathrm{\ell }+m}{B}_{2\mathrm{\ell }+1}^{2m+1}(h^2){\mathrm{Ms}}_{2m+1}^{(j)}(z,h),$$

ⓘ

Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $\sin z$: sine function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions, $\varphi (z,t)$: function, $R(z,t)$: function and $B_m(q)$: Fourier coefficient

A&S Ref:

20.7.27 (in different form and only for $\mathrm{\ell }=0$)

Referenced by:

Erratum (V1.0.11) for Equations (28.28.21) and (28.28.22)

Permalink:

http://dlmf.nist.gov/28.28.E22

Encodings:

pMML, png, TeX

Errata (effective with 1.0.11):

Originally the prefactor and uppper limit of integration were given incorrectly as ${\displaystyle \frac{2}{\pi }}{\int }_0^{\pi }$.

Reported 2015-05-20 by Ruslan Kabasayev

See also:

Annotations for §28.28(ii), §28.28 and Ch.28

…