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In applied mathematics, the Kelvin functions berν(x) and beiν(x) are the real and imaginary parts, respectively, of where x is real, and Jν(z), is the νth order Bessel function of the first kind. Similarly, the functions kerν(x) and keiν(x) are the real and imaginary parts, respectively, of where Kν(z) is the νth order modified Bessel function of the second kind. These functions are named after William Thomson, 1st Baron Kelvin. Below, Γ(z) is the gamma function and ψ(z) is the digamma function.

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  • Kelvin functions (en)
  • Funkcje Kelvina (pl)
  • Функции Кельвина (ru)
  • 开尔文函数 (zh)
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  • Funkcje Kelvina – funkcje powiązane z funkcjami Bessela zespolonego argumentu. Oznaczane są symbolami: gdzie jest zmienną zespoloną, a rzeczywisty parametr rzędem funkcji. (pl)
  • Функции Кельвина — группа бесселевых функций. Каждая их пара представляют решения дифференциального уравнения: Введены Уильямом Томсоном (лордом Кельвином), который исследовал их в приложениях. (ru)
  • 开尔文函数有两类，得名自開爾文勳爵。第一类 , 第二类 , (zh)
  • In applied mathematics, the Kelvin functions berν(x) and beiν(x) are the real and imaginary parts, respectively, of where x is real, and Jν(z), is the νth order Bessel function of the first kind. Similarly, the functions kerν(x) and keiν(x) are the real and imaginary parts, respectively, of where Kν(z) is the νth order modified Bessel function of the second kind. These functions are named after William Thomson, 1st Baron Kelvin. Below, Γ(z) is the gamma function and ψ(z) is the digamma function. (en)
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  • http://commons.wikimedia.org/wiki/Special:FilePath/Mplwp_Kelvin-bei-norm.svg
  • http://commons.wikimedia.org/wiki/Special:FilePath/Mplwp_Kelvin-bei.svg
  • http://commons.wikimedia.org/wiki/Special:FilePath/Mplwp_Kelvin-ber-norm.svg
  • http://commons.wikimedia.org/wiki/Special:FilePath/Mplwp_Kelvin-ber.svg
  • http://commons.wikimedia.org/wiki/Special:FilePath/Mplwp_Kelvin-kei-norm.svg
  • http://commons.wikimedia.org/wiki/Special:FilePath/Mplwp_Kelvin-kei.svg
  • http://commons.wikimedia.org/wiki/Special:FilePath/Mplwp_Kelvin-ker-norm.svg
  • http://commons.wikimedia.org/wiki/Special:FilePath/Mplwp_Kelvin-ker.svg
  • http://commons.wikimedia.org/wiki/Special:FilePath/Plot_of_the_Kelvin_function_bei(z)_in_the_complex_plane_from_-2-2i_to_2+2i_with_colors_created_with_Mathematica_13.1_function_ComplexPlot3D.svg
  • http://commons.wikimedia.org/wiki/Special:FilePath/Plot_of_the_Kelvin_function_ber(z)_in_the_complex_plane_from_-2-2i_to_2+2i_with_colors_created_with_Mathematica_13.1_function_ComplexPlot3D.svg
  • http://commons.wikimedia.org/wiki/Special:FilePath/Plot_of_the_Kelvin_function_kei(z)_in_the_complex_plane_from_-2-2i_to_2+2i_with_colors_created_with_Mathematica_13.1_function_ComplexPlot3D.svg
  • http://commons.wikimedia.org/wiki/Special:FilePath/Plot_of_the_Kelvin_function_ker(z)_in_the_complex_plane_from_-2-2i_to_2+2i_with_colors_created_with_Mathematica_13.1_function_ComplexPlot3D.svg
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  • F. W. J. (en)
  • L. C. (en)
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  • Olver (en)
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  • Bessel functions (en)
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  • In applied mathematics, the Kelvin functions berν(x) and beiν(x) are the real and imaginary parts, respectively, of where x is real, and Jν(z), is the νth order Bessel function of the first kind. Similarly, the functions kerν(x) and keiν(x) are the real and imaginary parts, respectively, of where Kν(z) is the νth 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 xeiφ, 0 ≤ φ < 2π. With the exception of bern(x) and bein(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. (en)
  • Funkcje Kelvina – funkcje powiązane z funkcjami Bessela zespolonego argumentu. Oznaczane są symbolami: gdzie jest zmienną zespoloną, a rzeczywisty parametr rzędem funkcji. (pl)
  • Функции Кельвина — группа бесселевых функций. Каждая их пара представляют решения дифференциального уравнения: Введены Уильямом Томсоном (лордом Кельвином), который исследовал их в приложениях. (ru)
  • 开尔文函数有两类，得名自開爾文勳爵。第一类 , 第二类 , (zh)
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