Kelvin ship-wave pattern

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11: 10.69 Uniform Asymptotic Expansions for Large Order

§10.69 Uniform Asymptotic Expansions for Large Order

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10.69.2 $${\mathrm{ber}}_{\nu }\left(\nu x\right)+\mathrm{i}{\mathrm{bei}}_{\nu }\left(\nu x\right)\sim \frac{{\mathrm{e}}^{\nu \xi }}{{(2\pi \nu \xi )}^{1/2}}{\left(\frac{x{\mathrm{e}}^{3\pi \mathrm{i}/4}}{1+\xi }\right)}^{\nu }\sum _{k=0}^{\mathrm{\infty }}\frac{U_k({\xi }^{-1})}{{\nu }^k},$$

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Symbols:

${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: 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:

9.10.37

Referenced by:

§10.69

Permalink:

http://dlmf.nist.gov/10.69.E2

Encodings:

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See also:

Annotations for §10.69 and Ch.10

… ►All fractional powers take their principal values. … ► … ►

12: 10.71 Integrals

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§10.71(i) Indefinite Integrals

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§10.71(ii) Definite Integrals

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§10.71(iii) Compendia

►For infinite double integrals involving Kelvin functions see Prudnikov et al. (1986b, pp. 630–631). … ►

13: 11.8 Analogs to Kelvin Functions

§11.8 Analogs to Kelvin Functions

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14: 10.66 Expansions in Series of Bessel Functions

§10.66 Expansions in Series of Bessel Functions

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10.66.1 $${\mathrm{ber}}_{\nu }x+\mathrm{i}{\mathrm{bei}}_{\nu }x=\sum _{k=0}^{\mathrm{\infty }}\frac{{\mathrm{e}}^{(3\nu +k)\pi \mathrm{i}/4}{x}^k{J}_{\nu +k}\left(x\right)}{2^{k/2}k!}=\sum _{k=0}^{\mathrm{\infty }}\frac{{\mathrm{e}}^{(3\nu +3k)\pi \mathrm{i}/4}{x}^k{I}_{\nu +k}\left(x\right)}{2^{k/2}k!}.$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, ${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $k$: nonnegative integer, $x$: real variable and $\nu$: complex parameter

A&S Ref:

9.9.32

Referenced by:

§10.66

Permalink:

http://dlmf.nist.gov/10.66.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.66 and Ch.10

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${\mathrm{ber}}_n\left(x\sqrt{2}\right)={\displaystyle \sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}}{(-1)}^{n+k}{J}_{n+2k}\left(x\right)I_{2k}\left(x\right),$

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${\mathrm{bei}}_n\left(x\sqrt{2}\right)={\displaystyle \sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}}{(-1)}^{n+k}{J}_{n+2k+1}\left(x\right)I_{2k+1}\left(x\right).$

15: 10.68 Modulus and Phase Functions

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§10.68(i) Definitions

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§10.68(ii) Basic Properties

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$M_{\nu }\left(x\right)={({\mathrm{ber}}_{\nu }^{2}x+{\mathrm{bei}}_{\nu }^{2}x)}^{1/2},$

… ►Equations (10.68.8)–(10.68.14) also hold with the symbols $\mathrm{ber}$, $\mathrm{bei}$, $M$, and $\theta$ replaced throughout by $\ker$, $\mathrm{kei}$, $N$, and $\varphi$, respectively. … ►

§10.68(iii) Asymptotic Expansions for Large Argument

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16: Sidebar 9.SB2: Interference Patterns in Caustics

Sidebar 9.SB2: Interference Patterns in Caustics

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17: Sidebar 21.SB1: Periodic Surface Waves

… ►The caption reads “Mosaic of two overhead photographs, showing surface patterns of waves in shallow water”. …

18: 10.76 Approximations

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Kelvin Functions

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19: Sidebar 7.SB1: Diffraction from a Straightedge

… ►The faint circular patterns are additional diffraction effects due to imperfections in the edge.

20: 10.1 Special Notation

… ►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 ${\mathrm{ber}}_{\nu }\left(x\right)$, ${\mathrm{bei}}_{\nu }\left(x\right)$, ${\ker }_{\nu }\left(x\right)$, ${\mathrm{kei}}_{\nu }\left(x\right)$. …For the Kelvin functions the order $\nu$ is always assumed to be real. … ►For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).