§10.64 Integral Representations

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Keywords:: Kelvin functions, integral representations
Permalink:: http://dlmf.nist.gov/10.64
See also:: Annotations for Ch.10

Schläfli-Type Integrals

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Keywords:: Kelvin functions, Schläfli-type integrals, derivatives, with respect to order
See also:: Annotations for §10.64 and Ch.10

10.64.1 $\operatorname{ber}_{n}\left(x\sqrt{2}\right)=\frac{(-1)^{n}}{\pi}\int_{0}^{\pi% }\cos\left(x\sin t-nt\right)\cosh\left(x\sin t\right)\,\mathrm{d}t,$

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Symbols:: $\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of , $\cosh\NVar{z}$ : hyperbolic cosine function, $\int$ : integral, $\sin\NVar{z}$ : sine function, : integer and : real variable
Permalink:: http://dlmf.nist.gov/10.64.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §10.64, §10.64 and Ch.10

10.64.2 $\operatorname{bei}_{n}\left(x\sqrt{2}\right)=\frac{(-1)^{n}}{\pi}\int_{0}^{\pi% }\sin\left(x\sin t-nt\right)\sinh\left(x\sin t\right)\,\mathrm{d}t.$

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Symbols:: $\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral, $\sin\NVar{z}$ : sine function, : integer and : real variable
Permalink:: http://dlmf.nist.gov/10.64.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §10.64, §10.64 and Ch.10

See Apelblat (1991) for these results, and also for similar representations for $\operatorname{ber}_{\nu}\left(x\sqrt{2}\right)$ , $\operatorname{bei}_{\nu}\left(x\sqrt{2}\right)$ , and their $\nu$ -derivatives.