small x

SearchAdvancedHelp

(0.005 seconds)

21—30 of 102 matching pages

21: 10.67 Asymptotic Expansions for Large Argument

… ►The contributions of the terms in ${\ker }_{\nu }x$, ${\mathrm{kei}}_{\nu }x$, ${\ker }_{\nu }^{\prime }x$, and ${\mathrm{kei}}_{\nu }^{\prime }x$ on the right-hand sides of (10.67.3), (10.67.4), (10.67.7), and (10.67.8) are exponentially small compared with the other terms, and hence can be neglected in the sense of Poincaré asymptotic expansions (§2.1(iii)). …

22: 9.1 Special Notation

… ► ►►►►►

$k$	nonnegative integer, except in §9.9(iii).
$x$	real variable.
$z(=x+\mathrm{i}y)$	complex variable.
$\delta$	arbitrary small positive constant.
…

… ►Other notations that have been used are as follows: $\mathrm{Ai}\left(-x\right)$ and $\mathrm{Bi}\left(-x\right)$ for $\mathrm{Ai}\left(x\right)$ and $\mathrm{Bi}\left(x\right)$ (Jeffreys (1928), later changed to $\mathrm{Ai}\left(x\right)$ and $\mathrm{Bi}\left(x\right)$); $U(x)=\sqrt{\pi }\mathrm{Bi}\left(x\right)$, $V(x)=\sqrt{\pi }\mathrm{Ai}\left(x\right)$ (Fock (1945)); $A(x)=3^{-1/3}\pi \mathrm{Ai}\left(-3^{-1/3}x\right)$ (Szegő (1967, §1.81)); $e_0(x)=\pi \mathrm{Hi}(-x)$, ${\tilde{e}}_0(x)=-\pi \mathrm{Gi}(-x)$ (Tumarkin (1959)).

23: 18.1 Notation

… ►

$x$-Differences

… ►

Legendre: $P_n\left(x\right)$.

… ►

Hermite: $H_n\left(x\right)$, ${\mathrm{𝐻𝑒}}_n\left(x\right)$.

… ►

Charlier: $C_n(x;a)$.

… ►

Bessel: $y_n(x;a)$.

…

24: 36.5 Stokes Sets

… ►Stokes sets are surfaces (codimension one) in $𝐱$ space, across which ${\mathrm{\Psi }}_K(𝐱;k)$ or ${\mathrm{\Psi }}^{(\mathrm{U})}(𝐱;k)$ acquires an exponentially-small asymptotic contribution (in $k$), associated with a complex critical point of ${\mathrm{\Phi }}_K$ or . …

25: 10.69 Uniform Asymptotic Expansions for Large Order

… ►Accuracy in (10.69.2) and (10.69.4) can be increased by including exponentially-small contributions as in (10.67.3), (10.67.4), (10.67.7), and (10.67.8) with $x$ replaced by $\nu x$. …

26: 19.12 Asymptotic Approximations

… ►With $\psi \left(x\right)$ denoting the digamma function (§5.2(i)) in this subsection, the asymptotic behavior of $K\left(k\right)$ and $E\left(k\right)$ near the singularity at $k=1$ is given by the following convergent series: … ►They are useful primarily when $(1-k)/(1-\sin \varphi )$ is either small or large compared with 1. ►If $x\ge 0$ and $y>0$, then ►

19.12.6 $$R_C(x,y)=\frac{\pi }{2\sqrt{y}}-\frac{\sqrt{x}}{y}\left(1+O\left(\sqrt{\frac{x}{y}}\right)\right),$$ $x/y\to 0$,

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions and $\pi$: the ratio of the circumference of a circle to its diameter

Referenced by:

§19.12

Permalink:

http://dlmf.nist.gov/19.12.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §19.12 and Ch.19

►

19.12.7 $$R_C(x,y)=\frac{1}{2\sqrt{x}}\left(\left(1+\frac{y}{2x}\right)\ln \left(\frac{4x}{y}\right)-\frac{y}{2x}\right)(1+O\left(y^2/x^2\right)),$$ $y/x\to 0$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions and $\ln z$: principal branch of logarithm function

Referenced by:

§19.12, §19.2(iv)

Permalink:

http://dlmf.nist.gov/19.12.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §19.12 and Ch.19

27: 18.26 Wilson Class: Continued

… ►

18.26.4 $$R_n(y(y+\gamma +\delta +1);\gamma ,\delta ,N)={}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{-n,-y,y+\gamma +\delta +1}{\gamma +1,-N};1),$$ $n=0,1,\mathrm{\dots },N$.

ⓘ

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $R_n(x;\gamma ,\delta ,N)$: dual Hahn polynomial, $y$: real variable, $N$: positive integer, $\delta$: arbitrary small positive constant and $n$: nonnegative integer

Referenced by:

§18.26(i), §18.26(ii)

Permalink:

http://dlmf.nist.gov/18.26.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §18.26(i), §18.26 and Ch.18

… ►

18.26.4_1 $$R_n(y(y+\gamma +\delta +1);\gamma ,\delta ,N)=Q_y(n;\gamma ,\delta ,N),$$

ⓘ

Symbols:

$Q_n(x;\alpha ,\beta ,N)$: Hahn polynomial, $R_n(x;\gamma ,\delta ,N)$: dual Hahn polynomial, $y$: real variable, $N$: positive integer, $\delta$: arbitrary small positive constant and $n$: nonnegative integer

Referenced by:

§18.26(i), Erratum (V1.2.0) Section 18.26

Permalink:

http://dlmf.nist.gov/18.26.E4_1

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.26(i), §18.26(i), §18.26 and Ch.18

… ►

18.26.4_2 $$R_n(y(y+\gamma +\delta +1);\alpha ,\beta ,\gamma ,\delta )=R_y(n(n+\alpha +\beta +1);\gamma ,\delta ,\alpha ,\beta ).$$

ⓘ

Symbols:

$R_n(x;\alpha ,\beta ,\gamma ,\delta )$: Racah polynomial, $y$: real variable, $\delta$: arbitrary small positive constant and $n$: nonnegative integer

Referenced by:

§18.26(i), Erratum (V1.2.0) Section 18.26

Permalink:

http://dlmf.nist.gov/18.26.E4_2

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.26(i), §18.26(i), §18.26 and Ch.18

… ►

18.26.9 $$\underset{\beta \to \mathrm{\infty }}{lim}{R}_n(x;-N-1,\beta ,\gamma ,\delta )=R_n(x;\gamma ,\delta ,N).$$

ⓘ

Symbols:

$R_n(x;\alpha ,\beta ,\gamma ,\delta )$: Racah polynomial, $R_n(x;\gamma ,\delta ,N)$: dual Hahn polynomial, $N$: positive integer, $\delta$: arbitrary small positive constant, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.26(ii)

Permalink:

http://dlmf.nist.gov/18.26.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §18.26(ii), §18.26(ii), §18.26 and Ch.18

… ►

18.26.10 $$\underset{\delta \to \mathrm{\infty }}{lim}{R}_n(x(x+\gamma +\delta +1);\alpha ,\beta ,-N-1,\delta )=Q_n(x;\alpha ,\beta ,N).$$

ⓘ

Symbols:

$Q_n(x;\alpha ,\beta ,N)$: Hahn polynomial, $R_n(x;\alpha ,\beta ,\gamma ,\delta )$: Racah polynomial, $N$: positive integer, $\delta$: arbitrary small positive constant, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.26(ii)

Permalink:

http://dlmf.nist.gov/18.26.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §18.26(ii), §18.26(ii), §18.26 and Ch.18

…

28: 6.1 Special Notation

… ► ►►►

$x$	real variable.
…
$\delta$	arbitrary small positive constant.
…

… ►The main functions treated in this chapter are the exponential integrals $\mathrm{Ei}\left(x\right)$, $E_1\left(z\right)$, and $\mathrm{Ein}\left(z\right)$; the logarithmic integral $\mathrm{li}\left(x\right)$; the sine integrals $\mathrm{Si}\left(z\right)$ and $\mathrm{si}\left(z\right)$; the cosine integrals $\mathrm{Ci}\left(z\right)$ and $\mathrm{Cin}\left(z\right)$. …

29: 18.24 Hahn Class: Asymptotic Approximations

… ►The first expansion holds uniformly for $\delta \le x\le 1+\delta$, and the second for $1-\delta \le x\le 1+{\delta }^{-1}$, $\delta$ being an arbitrary small positive constant. … ►Taken together, these expansions are uniformly valid for and for $a$ in unbounded intervals—each of which contains $[0,(1-\delta )n]$, where $\delta$ again denotes an arbitrary small positive constant. …

30: 2.1 Definitions and Elementary Properties

… ►As $x\to c$ in $𝐗$ … ►(Here and elsewhere in this chapter $\delta$ is an arbitrary small positive constant.) … ►If ${\sum }_{s=0}^{\mathrm{\infty }}{a}_s{z}^s$ converges for all sufficiently small $|z|$, then for each nonnegative integer $n$ … ►Then $\sum a_s{x}^{-s}$ is a Poincaré asymptotic expansion, or simply asymptotic expansion, of $f(x)$ as $x\to \mathrm{\infty }$ in $𝐗$. … ►Suppose also that $f(x)$ and $f_s(x)$ satisfy …