Kelvin functions

In applied mathematics, the Kelvin functions berν(x) and beiν(x) are the <a href="/facts/Real_part/2w7mqI7e">real</a> and <a href="/facts/Imaginary_part/2w7mqI7e">imaginary parts</a>, respectively, of

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    {\displaystyle J_{\nu }\left(xe^{\frac {3\pi i}{4}}\right),\,}

where x is real, and Jν(z), is the νth order <a href="/facts/Bessel_function/hSyvFPqC">Bessel function</a> of the first kind. Similarly, the functions kerν(x) and keiν(x) are the real and imaginary parts, respectively, of

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    {\displaystyle K_{\nu }\left(xe^{\frac {\pi i}{4}}\right),\,}

where Kν(z) is the νth order <a href="/facts/Bessel_function/hSyvFPqC">modified Bessel function</a> of the second kind.
These functions are named after <a href="/facts/William_Thomson,_1st_Baron_Kelvin/LTIxG1jV">William Thomson, 1st Baron Kelvin</a>.
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 <a href="/facts/Branch_point/lJ7Y6Iht">branch point</a> at x = 0.
Below, Γ(z) is the <a href="/facts/Gamma_function/SjGQcnyw">gamma function</a> and ψ(z) is the <a href="/facts/Digamma_function/nDTNu86e">digamma function</a>.

Kelvin functions open-in-new

Kelvin functions