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31: 19.36 Methods of Computation

… ►When the values of complete integrals are known, addition theorems with $\psi =\pi /2$ (§19.11(ii)) ease the computation of functions such as $F(\varphi ,k)$ when $\frac{1}{2}\pi -\varphi$ is small and positive. …

32: 18.28 Askey–Wilson Class

… ►

18.28.20 $$\sum _{y=0}^N{R}_n(q^{-y}+\gamma \delta q^{y+1})R_m(q^{-y}+\gamma \delta q^{y+1}){\omega }_y=h_n{\delta }_{n,m},$$ $n,m=0,1,\mathrm{\dots },N$,

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $y$: real variable, $N$: positive integer, $\delta$: arbitrary small positive constant, $q$: real variable, $m$: nonnegative integer, $n$: nonnegative integer, $h_n$ and $R_n(x)$

Permalink:

http://dlmf.nist.gov/18.28.E20

Encodings:

pMML, png, TeX

See also:

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

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33: 2.5 Mellin Transform Methods

… ►

§2.5(iii) Laplace Transforms with Small Parameters

… ►where $l$ ($\ge 2$) is an arbitrary integer and $\delta$ is an arbitrary small positive constant. … … ►To verify (2.5.48) we may use … ►For examples in which the integral defining the Mellin transform $ℳh\left(z\right)$ does not exist for any value of $z$, see Wong (1989, Chapter 3), Bleistein and Handelsman (1975, Chapter 4), and Handelsman and Lew (1970).

34: 10.69 Uniform Asymptotic Expansions for Large Order

… ►Let $U_k(p)$ and $V_k(p)$ be the polynomials defined in §10.41(ii), and … ►

10.69.3 $${\ker }_{\nu }\left(\nu x\right)+\mathrm{i}{\mathrm{kei}}_{\nu }\left(\nu x\right)\sim {\mathrm{e}}^{-\nu \xi }{\left(\frac{\pi }{2\nu \xi }\right)}^{1/2}{\left(\frac{x{\mathrm{e}}^{3\pi \mathrm{i}/4}}{1+\xi }\right)}^{-\nu }\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{U_k({\xi }^{-1})}{{\nu }^k},$$

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Symbols:

${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, ${\ker }_{\nu }\left(x\right)$: Kelvin function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $x$: real variable, $\nu$: complex parameter, $U_k(p)$: polynomial coefficient and $\xi$

A&S Ref:

9.10.38

Permalink:

http://dlmf.nist.gov/10.69.E3

Encodings:

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See also:

Annotations for §10.69 and Ch.10

… ►

10.69.5 $${\ker }_{\nu }^{\prime }\left(\nu x\right)+\mathrm{i}{\mathrm{kei}}_{\nu }^{\prime }\left(\nu x\right)\sim -\frac{{\mathrm{e}}^{-\nu \xi }}{x}{\left(\frac{\pi \xi }{2\nu }\right)}^{1/2}{\left(\frac{x{\mathrm{e}}^{3\pi \mathrm{i}/4}}{1+\xi }\right)}^{-\nu }\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{V_k({\xi }^{-1})}{{\nu }^k},$$

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Symbols:

${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, ${\ker }_{\nu }\left(x\right)$: Kelvin function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $x$: real variable, $\nu$: complex parameter, $V_k(p)$: polynomial coefficient and $\xi$

A&S Ref:

9.10.40

Permalink:

http://dlmf.nist.gov/10.69.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.69 and Ch.10

… ►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 $\nu x$.

35: 10.1 Special Notation

… ► ►►►►

$m,n$	integers. In §§10.47–10.71 $n$ is nonnegative.
$k$	nonnegative integer (except in §10.73).
…
$\delta$	arbitrary small positive constant.
…

►The main functions treated in this chapter are the Bessel functions $J_{\nu }\left(z\right)$, $Y_{\nu }\left(z\right)$; Hankel functions $H_{\nu }^{(1)}\left(z\right)$, $H_{\nu }^{(2)}\left(z\right)$; modified Bessel functions $I_{\nu }\left(z\right)$, $K_{\nu }\left(z\right)$; spherical Bessel functions ${𝗃}_n\left(z\right)$, ${𝗒}_n\left(z\right)$, ${𝗁}_n^{(1)}\left(z\right)$, ${𝗁}_n^{(2)}\left(z\right)$; modified spherical Bessel functions ${𝗂}_n^{(1)}\left(z\right)$, ${𝗂}_n^{(2)}\left(z\right)$, ${𝗄}_n\left(z\right)$; Kelvin functions ${\mathrm{ber}}_{\nu }\left(x\right)$, ${\mathrm{bei}}_{\nu }\left(x\right)$, ${\ker }_{\nu }\left(x\right)$, ${\mathrm{kei}}_{\nu }\left(x\right)$. … ►Jeffreys and Jeffreys (1956): ${\mathrm{Hs}}_{\nu }(z)$ for $H_{\nu }^{(1)}\left(z\right)$, ${\mathrm{Hi}}_{\nu }(z)$ for $H_{\nu }^{(2)}\left(z\right)$, ${\mathrm{Kh}}_{\nu }(z)$ for $(2/\pi )K_{\nu }\left(z\right)$. ►Whittaker and Watson (1927): $K_{\nu }\left(z\right)$ for $\cos \left(\nu \pi \right)K_{\nu }\left(z\right)$. …

36: 10.67 Asymptotic Expansions for Large Argument

… ►Define $a_k(\nu )$ and $b_k(\nu )$ as in §§10.17(i) and 10.17(ii). … ►

10.67.1 $${\ker }_{\nu }x\sim {\mathrm{e}}^{-x/\sqrt{2}}{\left(\frac{\pi }{2x}\right)}^{\frac{1}{2}}\sum _{k=0}^{\mathrm{\infty }}\frac{a_k(\nu )}{x^k}\cos \left(\frac{x}{\sqrt{2}}+\left(\frac{\nu }{2}+\frac{k}{4}+\frac{1}{8}\right)\pi \right),$$

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Symbols:

${\ker }_{\nu }\left(x\right)$: Kelvin function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $k$: nonnegative integer, $x$: real variable, $\nu$: complex parameter and $a_k(\nu )$: polynomial coefficient

A&S Ref:

9.10.3, 9.10.5--9.10.7 (modified)

Referenced by:

§10.61(v), §10.67(ii), §10.68(iii)

Permalink:

http://dlmf.nist.gov/10.67.E1

Encodings:

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See also:

Annotations for §10.67(i), §10.67 and Ch.10

►

10.67.2 $${\mathrm{kei}}_{\nu }x\sim -{\mathrm{e}}^{-x/\sqrt{2}}{\left(\frac{\pi }{2x}\right)}^{\frac{1}{2}}\sum _{k=0}^{\mathrm{\infty }}\frac{a_k(\nu )}{x^k}\sin \left(\frac{x}{\sqrt{2}}+\left(\frac{\nu }{2}+\frac{k}{4}+\frac{1}{8}\right)\pi \right).$$

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Symbols:

${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\sin z$: sine function, $k$: nonnegative integer, $x$: real variable, $\nu$: complex parameter and $a_k(\nu )$: polynomial coefficient

A&S Ref:

9.10.4, 9.10.5--9.10.7 (modified)

Referenced by:

§10.61(v)

Permalink:

http://dlmf.nist.gov/10.67.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.67(i), §10.67 and Ch.10

… ►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)). … …

37: 16.1 Special Notation

… ► ►►►►►►

$p,q$	nonnegative integers.
$k,n$	nonnegative integers, unless
…
$\delta$	arbitrary small positive constant.
…
${\left(𝐚\right)}_k$	${\left(a_1\right)}_k{\left(a_2\right)}_k\mathrm{\cdots }{\left(a_p\right)}_k$.
${\left(𝐛\right)}_k$	${\left(b_1\right)}_k{\left(b_2\right)}_k\mathrm{\cdots }{\left(b_q\right)}_k$.
…

…

38: 8.18 Asymptotic Expansions of $I_x(a,b)$

… ►The functions $F_k$ are defined by …The coefficients $d_k$ are defined by the generating function … ►A recurrence relation for the $d_k$ can be found in Nemes and Olde Daalhuis (2016). … ►uniformly for $x\in (0,1)$ and $a/(a+b)$, $b/(a+b)\in [\delta ,1-\delta ]$, where $\delta$ again denotes an arbitrary small positive constant. … ►For this result, and for higher coefficients $c_k(\eta )$ see Temme (1996b, §11.3.3.2). …

39: 5.1 Special Notation

… ► ►►►►

$j,m,n$	nonnegative integers.
$k$	nonnegative integer, except in §5.20.
…
$\delta$	arbitrary small positive constant.
…

…

40: 18.1 Notation

… ► ►►►►

$x,y,t$	real variables.
…
$k,l,\mathrm{\ell },$	nonnegative integers.
…
$\delta$	arbitrary small positive constant.
…

… ►

$${(z_1,\mathrm{\dots },z_k;q)}_n={(z_1;q)}_n\mathrm{\cdots }{(z_k;q)}_n.$$

… ►

$${(z_1,\mathrm{\dots },z_k;q)}_{\mathrm{\infty }}={(z_1;q)}_{\mathrm{\infty }}\mathrm{\cdots }{(z_k;q)}_{\mathrm{\infty }}.$$

… ►

Krawtchouk: $K_n(x;p,N)$.

…