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11—20 of 22 matching pages

11: 18.24 Hahn Class: Asymptotic Approximations

… ►With

x=\lambda N

and

\nu=n/N

, Li and Wong (2000) gives an asymptotic expansion for

K_{n}\left(x;p,N\right)

n\to\infty

, that holds uniformly for

\lambda

and

\nu

in compact subintervals of

(0,1)

. …

12: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►For

\mathcal{D}(T)

we can take

C^{2}(X)

, with appropriate boundary conditions, and with compact support if

X

is bounded, which space is dense in

L^{2}\left(X\right)

, and for

X

unbounded require that possible non- $L^{2}$ eigenfunctions of (1.18.28), with real eigenvalues, are non-zero but bounded on open intervals, including

\pm\infty

. …

13: 10.43 Integrals

… ►where

\psi=\ifrac{\Gamma'}{\Gamma}

and

\gamma

is Euler’s constant (§5.2). … ►provided that either of the following sets of conditions is satisfied: ►

(a)

On the interval $0<x<\infty$ , $x^{-1}g(x)$ is continuously differentiable and each of $xg(x)$ and $x\ifrac{\mathrm{d}(x^{-1}g(x))}{\mathrm{d}x}$ is absolutely integrable.

►

(b)

$g(x)$ is piecewise continuous and of bounded variation on every compact interval in $(0,\infty)$ , and each of the following integrals

…

14: 24.11 Asymptotic Approximations

… ►

24.11.5

(-1)^{\left\lfloor n/2\right\rfloor-1}\frac{(2\pi)^{n}}{2(n!)}B_{n}\left(x% \right)\to\begin{cases}\cos\left(2\pi x\right),&n\text{ even},\\ \sin\left(2\pi x\right),&n\text{ odd},\end{cases}

ⓘ

Symbols:: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $!$ : factorial (as in $n!$ ), $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $\sin\NVar{z}$ : sine function, $n$ : integer and $x$ : real or complex
Referenced by:: §24.11
Permalink:: http://dlmf.nist.gov/24.11.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §24.11 and Ch.24

►

24.11.6

(-1)^{\left\lfloor(n+1)/2\right\rfloor}\frac{\pi^{n+1}}{4(n!)}E_{n}\left(x% \right)\to\begin{cases}\sin\left(\pi x\right),&n\text{ even},\\ \cos\left(\pi x\right),&n\text{ odd},\end{cases}

ⓘ

Symbols:: $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $!$ : factorial (as in $n!$ ), $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $\sin\NVar{z}$ : sine function, $n$ : integer and $x$ : real or complex
Referenced by:: §24.11
Permalink:: http://dlmf.nist.gov/24.11.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §24.11 and Ch.24

►uniformly for

x

on compact subsets of

\mathbb{C}

. …

15: 28.2 Definitions and Basic Properties

… ►converges absolutely and uniformly in compact subsets of

\mathbb{C}

. … ►

28.2.19

{qc_{2n+2}-\left(a-(\nu+2n)^{2}\right)c_{2n}+qc_{2n-2}=0,}

n\in\mathbb{Z}

ⓘ

Symbols:: $\in$ : element of, $\mathbb{Z}$ : set of all integers, $q=h^{2}$ : parameter, $n$ : integer, $\nu$ : complex parameter, $a$ : parameter and $c_{2n}$ : coefficients
A&S Ref:: 20.3.12 20.3.13
Referenced by:: §28.2(iv)
Permalink:: http://dlmf.nist.gov/28.2.E19
Encodings:: pMML, png, TeX
See also:: Annotations for §28.2(iv), §28.2 and Ch.28

… ►For given

\nu

and

q

, equation (28.2.16) determines an infinite discrete set of values of

a

, the eigenvalues or characteristic values, of Mathieu’s equation. When

\widehat{\nu}=0

1

, the notation for the two sets of eigenvalues corresponding to each

\widehat{\nu}

is shown in Table 28.2.1, together with the boundary conditions of the associated eigenvalue problem. … ►

\operatorname{ce}_{0}\left(z,0\right)=1/\sqrt{2},

…

DiDonato (1978) gives a simple approximation for the function $F(p,x)=x^{-p}e^{x^{2}/2}\int_{x}^{\infty}e^{-t^{2}/2}t^{p}\,\mathrm{d}t$ (which is related to the incomplete gamma function by a change of variables) for real $p$ and large positive $x$ . This takes the form $F(p,x)=4x/h(p,x)$ , approximately, where $h(p,x)=3(x^{2}-p)+\sqrt{(x^{2}-p)^{2}+8(x^{2}+p)}$ and is shown to produce an absolute error $O\left(x^{-7}\right)$ as $x\to\infty$ .

… ►

•

Luke (1969b, p. 186) gives hypergeometric polynomial representations that converge uniformly on compact subsets of the $z$ -plane that exclude $z=0$ and are valid for $\left|\operatorname{ph}z\right|<\pi$ .

… ►

•

Verbeeck (1970) gives polynomial and rational approximations for $E_{p}\left(x\right)=(e^{-x}/x)P(z)$ , approximately, where $P(z)$ denotes a quotient of polynomials of equal degree in $z=x^{-1}$ .

18: 18.15 Asymptotic Approximations

… ►For large

\beta

, fixed

\alpha

, and

0\leq n/\beta\leq c

, Dunster (1999) gives asymptotic expansions of

P^{(\alpha,\beta)}_{n}\left(z\right)

that are uniform in unbounded complex

z

-domains containing

z=\pm 1

. …This reference also supplies asymptotic expansions of

P^{(\alpha,\beta)}_{n}\left(z\right)

for large

n

, fixed

\alpha

, and

0\leq\beta/n\leq c

. … ►Asymptotic expansions for

C^{(\lambda)}_{n}\left(\cos\theta\right)

can be obtained from the results given in §18.15(i) by setting

\alpha=\beta=\lambda-\frac{1}{2}

and referring to (18.7.1). … ►as

n\to\infty

, uniformly on compact

x

-intervals in

(0,\infty)

, where … ►as

n\to\infty

, uniformly on compact

x

-intervals on

\mathbb{R}

. …

19: 13.8 Asymptotic Approximations for Large Parameters

… ►

§13.8(ii) Large $b$ and $z$ , Fixed $a$ and $b/z$

►Let

\lambda=z/b>0

and

\zeta=\sqrt{2(\lambda-1-\ln\lambda)}

with

\operatorname{sign}\left(\zeta\right)=\operatorname{sign}\left(\lambda-1\right)

. … ►as

b\to\infty

, uniformly in compact

\lambda

-intervals of

(0,\infty)

and compact real

a

-intervals. … ►where

w=\operatorname{arccosh}\left(1+(2a)^{-1}x\right)

, and

\beta=\ifrac{(w+\sinh w)}{2}

. … ►For asymptotic approximations to

M\left(a,b,x\right)

and

U\left(a,b,x\right)

a\to-\infty

that hold uniformly with respect to

x\in(0,\infty)

and bounded positive values of

(b-1)/\left|a\right|

, combine (13.14.4), (13.14.5) with §§13.21(ii), 13.21(iii). …

20: 28.32 Mathematical Applications

… ►This leads to integral equations and an integral relation between the solutions of Mathieu’s equation (setting

\zeta=\mathrm{i}\xi

z=\eta

in (28.32.3)). … ►defines a solution of Mathieu’s equation, provided that (in the case of an improper curve) the integral converges with respect to

z

uniformly on compact subsets of

\mathbb{C}

. …

compact set

11—20 of 22 matching pages

11: 18.24 Hahn Class: Asymptotic Approximations

12: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

13: 10.43 Integrals

14: 24.11 Asymptotic Approximations

15: 28.2 Definitions and Basic Properties

16: 18.18 Sums

17: 8.27 Approximations

18: 18.15 Asymptotic Approximations

19: 13.8 Asymptotic Approximations for Large Parameters

§13.8(ii) Large $b$ and $z$ , Fixed $a$ and $b/z$

20: 28.32 Mathematical Applications

compact set

11—20 of 22 matching pages

11: 18.24 Hahn Class: Asymptotic Approximations

12: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

13: 10.43 Integrals

14: 24.11 Asymptotic Approximations

15: 28.2 Definitions and Basic Properties

16: 18.18 Sums

17: 8.27 Approximations

18: 18.15 Asymptotic Approximations

19: 13.8 Asymptotic Approximations for Large Parameters

§13.8(ii) Large b and z , Fixed a and b / z

20: 28.32 Mathematical Applications

§13.8(ii) Large $b$ and $z$ , Fixed $a$ and $b/z$