开尔文函数

开尔文函数有两类^[1]，得名自开尔文勋爵。 第一类 ${\displaystyle ber_{v}(x)}$,${\displaystyle bei_{v}(x)}$

第二类 ${\displaystyle ker_{v}(x)}$,${\displaystyle kei_{v}(x)}$

第一类开尔文函数

Kelvin Ber(v,z) function

Kelvin Bei(v,z) function

${\displaystyle ber_{v}(x)=Re(J_{\nu }\left(xe^{\frac {3\pi i}{4}}\right))}$

其中${\displaystyle Re(f)}$代表${\displaystyle f}$的实数部分，${\displaystyle J_{\nu }}$是第一类贝塞尔函数。

${\displaystyle ber_{v}(x)=Im(J_{\nu }\left(xe^{\frac {3\pi i}{4}}\right))}$

其中${\displaystyle Im(f)}$代表${\displaystyle f}$的虚数部分。

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

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

其中n为整数，上式分母的${\displaystyle \Gamma }$是Γ函数。

第二类开尔文函数

Kelvin Ker(v,z) function

Kelvin Kei(v,z) function

${\displaystyle Ker_{v}(x)=Re(K_{\nu }\left(xe^{\frac {\pi i}{4}}\right))}$

${\displaystyle Kei_{v}(x)=Im(K_{\nu }\left(xe^{\frac {\pi i}{4}}\right))}$ 其中${\displaystyle K_{\nu }}$是第二类修正贝塞尔函数。

${\displaystyle {\begin{aligned}\mathrm {ker} _{n}(x)&=-\ln \left({\frac {x}{2}}\right)\mathrm {ber} _{n}(x)+{\frac {\pi }{4}}\mathrm {bei} _{n}(x)\\&\qquad +{\frac {1}{2}}\left({\frac {x}{2}}\right)^{-n}\sum _{k=0}^{n-1}\cos \left[\left({\frac {3n}{4}}+{\frac {k}{2}}\right)\pi \right]{\frac {(n-k-1)!}{k!}}\left({\frac {x^{2}}{4}}\right)^{k}\\&\qquad \qquad +{\frac {1}{2}}\left({\frac {x}{2}}\right)^{n}\sum _{k\geq 0}\cos \left[\left({\frac {3n}{4}}+{\frac {k}{2}}\right)\pi \right]{\frac {\psi (k+1)+\psi (n+k+1)}{k!(n+k)!}}\left({\frac {x^{2}}{4}}\right)^{k}\end{aligned}}}$

${\displaystyle {\begin{aligned}\mathrm {kei} _{n}(x)&=-\ln \left({\frac {x}{2}}\right)\mathrm {bei} _{n}(x)-{\frac {\pi }{4}}\mathrm {ber} _{n}(x)\\&\qquad -{\frac {1}{2}}\left({\frac {x}{2}}\right)^{-n}\sum _{k=0}^{n-1}\sin \left[\left({\frac {3n}{4}}+{\frac {k}{2}}\right)\pi \right]{\frac {(n-k-1)!}{k!}}\left({\frac {x^{2}}{4}}\right)^{k}\\&\qquad \qquad +{\frac {1}{2}}\left({\frac {x}{2}}\right)^{n}\sum _{k\geq 0}\sin \left[\left({\frac {3n}{4}}+{\frac {k}{2}}\right)\pi \right]{\frac {\psi (k+1)+\psi (n+k+1)}{k!(n+k)!}}\left({\frac {x^{2}}{4}}\right)^{k}\end{aligned}}}$

上式的${\displaystyle \psi }$是双伽玛函数。