Kelvin functions

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1: 11.8 Analogs to Kelvin Functions

§11.8 Analogs to Kelvin Functions

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2: 10.64 Integral Representations

§10.64 Integral Representations

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Schläfli-Type Integrals

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10.64.1 $${\mathrm{ber}}_n\left(x\sqrt{2}\right)=\frac{{(-1)}^n}{\pi }{\int }_0^{\pi }\cos \left(x\sin t-nt\right)\cosh \left(x\sin t\right)dt,$$

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Symbols:

${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\int$: integral, $\sin z$: sine function, $n$: integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/10.64.E1

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See also:

Annotations for §10.64, §10.64 and Ch.10

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10.64.2 $${\mathrm{bei}}_n\left(x\sqrt{2}\right)=\frac{{(-1)}^n}{\pi }{\int }_0^{\pi }\sin \left(x\sin t-nt\right)\sinh \left(x\sin t\right)dt.$$

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Symbols:

${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\sinh z$: hyperbolic sine function, $\int$: integral, $\sin z$: sine function, $n$: integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/10.64.E2

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See also:

Annotations for §10.64, §10.64 and Ch.10

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3: 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). … ►

4: 10.70 Zeros

§10.70 Zeros

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5: 10.62 Graphs

§10.62 Graphs

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6: 10.61 Definitions and Basic Properties

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§10.61(i) Definitions

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§10.61(ii) Differential Equations

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§10.61(iii) Reflection Formulas for Arguments

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§10.61(iv) Reflection Formulas for Orders

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§10.61(v) Orders $\pm \frac{1}{2}$

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7: 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

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Annotations for §10.69 and Ch.10

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

8: 10.65 Power Series

§10.65 Power Series

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§10.65(iii) Cross-Products and Sums of Squares

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10.65.6 $${\mathrm{ber}}_{\nu }^{2}x+{\mathrm{bei}}_{\nu }^{2}x={(\frac{1}{2}x)}^{2\nu }\sum _{k=0}^{\mathrm{\infty }}\frac{1}{\mathrm{\Gamma }\left(\nu +k+1\right)\mathrm{\Gamma }\left(\nu +2k+1\right)}\frac{{(\frac{1}{4}{x}^2)}^{2k}}{k!},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, $!$: factorial (as in $n!$), $k$: nonnegative integer, $x$: real variable and $\nu$: complex parameter

A&S Ref:

9.9.28

Permalink:

http://dlmf.nist.gov/10.65.E6

Encodings:

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See also:

Annotations for §10.65(iii), §10.65 and Ch.10

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§10.65(iv) Compendia

►For further power series summable in terms of Kelvin functions and their derivatives see Hansen (1975).

9: 10.63 Recurrence Relations and Derivatives

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

►Let $f_{\nu }(x)$, $g_{\nu }(x)$ denote any one of the ordered pairs: … ►

$\sqrt{2}{\mathrm{kei}}^{\prime }x=-{\ker }_{1}x+{\mathrm{kei}}_{1}x.$

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§10.63(ii) Cross-Products

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10: 10.67 Asymptotic Expansions for Large Argument

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

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§10.67(ii) Cross-Products and Sums of Squares in the Case $\nu =0$

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10.67.9 $${\mathrm{ber}}^{2}x+{\mathrm{bei}}^{2}x\sim \frac{{\mathrm{e}}^{x\sqrt{2}}}{2\pi x}\left(1+\frac{1}{4\sqrt{2}}\frac{1}{x}+\frac{1}{64}\frac{1}{x^2}-\frac{33}{256\sqrt{2}}\frac{1}{x^3}-\frac{1797}{8192}\frac{1}{x^4}+\mathrm{\cdots }\right),$$

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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 and $x$: real variable

A&S Ref:

9.10.27

Permalink:

http://dlmf.nist.gov/10.67.E9

Encodings:

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See also:

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

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10.67.10 $$\mathrm{ber}x{\mathrm{bei}}^{\prime }x-{\mathrm{ber}}^{\prime }x\mathrm{bei}x\sim \frac{{\mathrm{e}}^{x\sqrt{2}}}{2\pi x}\left(\frac{1}{\sqrt{2}}+\frac{1}{8}\frac{1}{x}+\frac{9}{64\sqrt{2}}\frac{1}{x^2}+\frac{39}{512}\frac{1}{x^3}+\frac{75}{8192\sqrt{2}}\frac{1}{x^4}+\mathrm{\cdots }\right),$$

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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 and $x$: real variable

A&S Ref:

9.10.28

Permalink:

http://dlmf.nist.gov/10.67.E10

Encodings:

pMML, png, TeX

See also:

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

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