beij%C3%for%20kernel

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11: 10.67 Asymptotic Expansions for Large Argument

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§10.67(i) $\operatorname{ber}_{\nu}x,\operatorname{bei}_{\nu}x,\operatorname{ker}_{\nu}x,% \operatorname{kei}_{\nu}x$ , and Derivatives

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10.67.1

\operatorname{ker}_{\nu}x\sim e^{-x/\sqrt{2}}\left(\frac{\pi}{2x}\right)^{% \frac{1}{2}}\*\sum_{k=0}^{\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:: $\operatorname{ker}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : 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:: pMML, png, TeX
See also:: Annotations for §10.67(i), §10.67 and Ch.10

… ►The contributions of the terms in

\operatorname{ker}_{\nu}x

\operatorname{kei}_{\nu}x

\operatorname{ker}_{\nu}'x

, and

\operatorname{kei}_{\nu}'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)). … ►

10.67.14

\operatorname{ker}x\operatorname{kei}'x-\operatorname{ker}'x\operatorname{kei}% x\sim-\frac{\pi}{2x}e^{-x\sqrt{2}}\left(\frac{1}{\sqrt{2}}-\frac{1}{8}\frac{1}% {x}+\frac{9}{64\sqrt{2}}\frac{1}{x^{2}}-\frac{39}{512}\frac{1}{x^{3}}+\frac{75% }{8192\sqrt{2}}\frac{1}{x^{4}}+\dotsb\right),

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Symbols:: $\operatorname{kei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ker}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm and $x$ : real variable
A&S Ref:: 9.10.32
Permalink:: http://dlmf.nist.gov/10.67.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §10.67(ii), §10.67 and Ch.10

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10.67.15

\operatorname{ker}x\operatorname{ker}'x+\operatorname{kei}x\operatorname{kei}'% x\sim-\frac{\pi}{2x}e^{-x\sqrt{2}}\left(\frac{1}{\sqrt{2}}+\frac{3}{8}\frac{1}% {x}-\frac{15}{64\sqrt{2}}\frac{1}{x^{2}}+\frac{45}{512}\frac{1}{x^{3}}+\frac{3% 15}{8192\sqrt{2}}\frac{1}{x^{4}}+\dotsb\right),

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Symbols:: $\operatorname{kei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ker}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm and $x$ : real variable
A&S Ref:: 9.10.33
Permalink:: http://dlmf.nist.gov/10.67.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §10.67(ii), §10.67 and Ch.10

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12: 10.73 Physical Applications

… ►See Krivoshlykov (1994, Chapter 2, §2.2.10; Chapter 5, §5.2.2), Kapany and Burke (1972, Chapters 4–6; Chapter 7, §A.1), and Slater (1942, Chapter 4, §§20, 25). … ►The analysis of the current distribution in circular conductors leads to the Kelvin functions

\operatorname{ber}x

\operatorname{bei}x

\operatorname{ker}x

, and

\operatorname{kei}x

. …

13: 8.26 Tables

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Khamis (1965) tabulates $P\left(a,x\right)$ for $a=0.05(.05)10(.1)20(.25)70$ , $0.0001\leq x\leq 250$ to 10D.

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Abramowitz and Stegun (1964, pp. 245–248) tabulates $E_{n}\left(x\right)$ for $n=2,3,4,10,20$ , $x=0(.01)2$ to 7D; also $(x+n)e^{x}E_{n}\left(x\right)$ for $n=2,3,4,10,20$ , $x^{-1}=0(.01)0.1(.05)0.5$ to 6S.

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Pagurova (1961) tabulates $E_{n}\left(x\right)$ for $n=0(1)20$ , $x=0(.01)2(.1)10$ to 4-9S; $e^{x}E_{n}\left(x\right)$ for $n=2(1)10$ , $x=10(.1)20$ to 7D; $e^{x}E_{p}\left(x\right)$ for $p=0(.1)1$ , $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

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Zhang and Jin (1996, Table 19.1) tabulates $E_{n}\left(x\right)$ for $n=1,2,3,5,10,15,20$ , $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.

14: 23 Weierstrass Elliptic and Modular
Functions

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15: 10.63 Recurrence Relations and Derivatives

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§10.63(i) $\operatorname{ber}_{\nu}x$ , $\operatorname{bei}_{\nu}x$ , $\operatorname{ker}_{\nu}x$ , $\operatorname{kei}_{\nu}x$

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\operatorname{ker}_{\nu}x,\operatorname{kei}_{\nu}x;

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\operatorname{kei}_{\nu}x,-\operatorname{ker}_{\nu}x.

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\sqrt{2}\operatorname{ker}'x=\operatorname{ker}_{1}x+\operatorname{kei}_{1}x,

… ►Equations (10.63.6) and (10.63.7) also hold when the symbols

\operatorname{ber}

and

\operatorname{bei}

in (10.63.5) are replaced throughout by

\operatorname{ker}

and

\operatorname{kei}

, respectively. …

16: 12.16 Mathematical Applications

… ►PCFs are also used in integral transforms with respect to the parameter, and inversion formulas exist for kernels containing PCFs. …

17: 36 Integrals with Coalescing Saddles

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18: Gergő Nemes

… ►As of September 20, 2021, Nemes performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 25 Zeta and Related Functions. …

19: Wolter Groenevelt

… ►As of September 20, 2022, Groenevelt performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 18 Orthogonal Polynomials. …

20: 33.24 Tables

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Abramowitz and Stegun (1964, Chapter 14) tabulates $F_{0}\left(\eta,\rho\right)$ , $G_{0}\left(\eta,\rho\right)$ , $F_{0}'\left(\eta,\rho\right)$ , and $G_{0}'\left(\eta,\rho\right)$ for $\eta=0.5(.5)20$ and $\rho=1(1)20$ , 5S; $C_{0}\left(\eta\right)$ for $\eta=0(.05)3$ , 6S.

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