开尔文函数

开尔文函数有两类，得名自開爾文勳爵。 第一类 b e r v ( x ) {\displaystyle ber_{v}(x)} , b e i v ( x ) {\displaystyle bei_{v}(x)} 第二类 k e r v ( x ) {\displaystyle ker_{v}(x)} , k e i v ( x ) {\displaystyle kei_{v}(x)} b e r v ( x ) = R e ( J ν ( x e 3 π i 4 ) ) {\displaystyle ber_{v}(x)=Re(J_{\nu }\left(xe^{\frac {3\pi i}{4}}\right))} 其中 R e ( f ) {\displaystyle Re(f)} 代表 f {\displaystyle f} 的实数部分， J ν {\displaystyle J_{\nu }} 是第一类贝塞尔函数。 b e r v ( x ) = I m ( J ν ( x e 3 π i 4 ) ) {\displaystyle ber_{v}(x)=Im(J_{\nu }\left(xe^{\frac {3\pi i}{4}}\right))} 其中 I m ( f ) {\displaystyle Im(f)} 代表 f {\displaystyle f} 的虚数部分。 b e r n ( x ) = ( x 2 ) n ∑ k ≥ 0 cos ⁡ [ ( 3 n 4 + k 2 ) π ] k ! Γ ( n + k + 1 ) ( x 2 4 ) k {\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}} b e i n ( x ) = ( x 2 ) n ∑ k ≥ 0 sin ⁡ [ ( 3 n 4 + k 2 ) π ] k ! Γ ( n + k + 1 ) ( x 2 4 ) 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 Γ {\displaystyle \Gamma } 是Γ函数。 K e r v ( x ) = R e ( K ν ( x e π i 4 ) ) {\displaystyle Ker_{v}(x)=Re(K_{\nu }\left(xe^{\frac {\pi i}{4}}\right))} K e i v ( x ) = I m ( K ν ( x e π i 4 ) ) {\displaystyle Kei_{v}(x)=Im(K_{\nu }\left(xe^{\frac {\pi i}{4}}\right))} 其中 K ν {\displaystyle K_{\nu }} 是第二類修正贝塞尔函数。 k e r n ( x ) = − ln ⁡ ( x 2 ) b e r n ( x ) + π 4 b e i n ( x ) + 1 2 ( x 2 ) − n ∑ k = 0 n − 1 cos ⁡ [ ( 3 n 4 + k 2 ) π ] ( n − k − 1 ) ! k ! ( x 2 4 ) k + 1 2 ( x 2 ) n ∑ k ≥ 0 cos ⁡ [ ( 3 n 4 + k 2 ) π ] ψ ( k + 1 ) + ψ ( n + k + 1 ) k ! ( n + k ) ! ( x 2 4 ) k {\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}}} k e i n ( x ) = − ln ⁡ ( x 2 ) b e i n ( x ) − π 4 b e r n ( x ) − 1 2 ( x 2 ) − n ∑ k = 0 n − 1 sin ⁡ [ ( 3 n 4 + k 2 ) π ] ( n − k − 1 ) ! k ! ( x 2 4 ) k + 1 2 ( x 2 ) n ∑ k ≥ 0 sin ⁡ [ ( 3 n 4 + k 2 ) π ] ψ ( k + 1 ) + ψ ( n + k + 1 ) k ! ( n + k ) ! ( x 2 4 ) k {\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 ψ {\displaystyle \psi } 是双伽玛函数。 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）