Kelvin functions

The Kelvin functions Ber$_{n}(x)$ and Bei$_{n}(x)$ are the real and imaginary parts, respectively, of $J_{n}(xe^{3\pi i/4})$, where $x$ is real, and $J_{n}(z)$ is the $n$^th order Bessel function of the first kind. Similarly, the functions Ker$_{n}(x)$ and Kei$_{n}(x)$ are the real and imaginary parts, respectively, of $K_{n}(xe^{3\pi i/4})$, $K_{n}(z)$ is the $n$^th order modified Bessel function of the second kind.

Ber$_{n}(x)$

Ber$_{n}(x)$ has the series expansion

$\mathrm {Ber} _{n}(x)=\left({\frac {x}{2}}\right)^{n}\sum _{k\geq 0}{\frac {\cos \left[\left({\frac {3n}{4}}+{\frac {k}{2}}\right)\pi \right]}{k!\Gamma (n+k+1)}}\left({\frac {x^{2}}{4}}\right)^{k}$

where $\Gamma (z)$ is the Gamma function. The special case Ber$_{0}(x)$, commonly denoted as just Ber$(x)$, has the series expansion

$\mathrm {Ber} (x)=1+\sum _{k\geq 1}{\frac {(-1)^{k}(x/2)^{4k}}{[(2k)!]^{2}}}$

and asymptotic series

$\mathrm {Ber} (x)\sim {\frac {e^{\frac {x}{\sqrt {2}}}}{\sqrt {2\pi x}}}[f_{1}(x)\cos \alpha +g_{1}(x)\sin \alpha ]-{\frac {\mathrm {Kei} (x)}{\pi }}$,

where $\alpha =x/{\sqrt {2}}-\pi /8$, and

$f_{1}(x)=1+\sum _{k\geq 1}{\frac {\cos(k\pi /4)}{k!(8x)^{k}}}\prod _{l=1}^{k}(2l-1)^{2}$

$g_{1}(x)=\sum _{k\geq 1}{\frac {\sin(k\pi /4)}{k!(8x)^{k}}}\prod _{l=1}^{k}(2l-1)^{2}$

Bei$_{n}(x)$

Bei$_{n}(x)$ has the series expansion

$\mathrm {Bei} _{n}(x)=\left({\frac {x}{2}}\right)^{n}\sum _{k\geq 0}{\frac {\sin \left[\left({\frac {3n}{4}}+{\frac {k}{2}}\right)\pi \right]}{k!\Gamma (n+k+1)}}\left({\frac {x^{2}}{4}}\right)^{k}$

where $\Gamma (z)$ is the Gamma function. The special case Bei$_{0}(x)$, commonly denoted as just Bei$(x)$, has the series expansion

$\mathrm {Bei} (x)=\sum _{k\geq 0}{\frac {(-1)^{k}(x/2)^{4k+2}}{[(2k+1)!]^{2}}}$

and asymptotic series

$\mathrm {Bei} (x)\sim {\frac {e^{\frac {x}{\sqrt {2}}}}{\sqrt {2\pi x}}}[f_{1}(x)\sin \alpha +g_{1}(x)\cos \alpha ]-{\frac {\mathrm {Ker} (x)}{\pi }}$,

where $\alpha$, $f_{1}(x)$, and $g_{1}(x)$ are defined as for Ber$(x)$.