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mathematical physics > special function > hypergeometric function > Kelvin function

mathematical analysis > functional analysis > special function > hypergeometric function > Kelvin function

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Kelvin function

In applied mathematics, the Kelvin functions ber_ν(x) and bei_ν(x) are the real and imaginary parts, respectively, of

${\\displaystyle J_{\ u }\\left(xe^{\ rac {3\\pi i}{4}}\ ight),\\,}$

where x is real, and $J ν (z)$ , is the ν^th order Bessel function of the first kind. Similarly, the functions ker_ν(x) and kei_ν(x) are the real and imaginary parts, respectively, of

${\\displaystyle K_{\ u }\\left(xe^{\ rac {\\pi i}{4}}\ ight),\\,}$

where $K ν (z)$ is the ν^th order modified Bessel function of the second kind.
These functions are named after William Thomson, 1st Baron Kelvin.
While the Kelvin functions are defined as the real and imaginary parts of Bessel functions with x taken to be real, the functions can be analytically continued for complex arguments $xe iφ, 0 \leq φ < 2 π .$ With the exception of ber_n(x) and bei_n(x) for integral n, the Kelvin functions have a branch point at x = 0.
Below, $Γ(z)$ is the gamma function and $ψ (z)$ is the digamma function.

(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Kelvin_functions)

hypergeometric function

fonction de Kelvin-Bessel

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http://data.loterre.fr/ark:/67375/PSR-DD268JMV-3

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RDF/XML TURTLE JSON-LD Date de création 17/08/2023, dernière modification le 17/08/2023