10 Bessel Functions Kelvin Functions 10.63 Recurrence Relations and Derivatives 10.65 Power Series

§10.64 Integral Representations

Keywords:: Kelvin functions
Permalink:: http://dlmf.nist.gov/10.64

¶ Schläfli-Type Integrals

Keywords:: Kelvin functions, Schläfli-type integrals, derivatives

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

Symbols:: $\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, : integer and : real variable
Permalink:: http://dlmf.nist.gov/10.64.E1
Encodings:: TeX, pMML, png

10.64.2 $\mathop{\mathrm{bei}_{{n}}\/}\nolimits\!\left(x\sqrt{2}\right)=\frac{(-1)^{n}}% {\pi}\int_{0}^{\pi}\mathop{\sin\/}\nolimits(x\mathop{\sin\/}\nolimits t-nt)% \mathop{\sinh\/}\nolimits(x\mathop{\sin\/}\nolimits t)dt.$

Symbols:: $\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, : differential of , $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, : integer and : real variable
Permalink:: http://dlmf.nist.gov/10.64.E2
Encodings:: TeX, pMML, png

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

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