开尔文函数

开尔文函数有两类^[1]，得名自開爾文勳爵。 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${\displaystyle K_{\nu }}$ 是第二類修正贝塞尔函数。

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

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

上式的${\displaystyle \psi }$ 是双伽玛函数。

参考文献

1. NIST HANDBOOK p261-268

• Olver, Frank W. J.; Lozier, Daniel W.; Boisvert, Ronald F.; Clark, Charles W. (编), NIST Handbook of Mathematical Functions, Cambridge University Press, 2010 [2015-01-24], ISBN 978-0-521-19225-5, MR 2723248, （原始内容存档于2013-07-03）