查看“︁开尔文函数”︁的源代码

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'''开尔文函数'''有两类<ref>NIST HANDBOOK p261-268</ref>，得名自[[第一代開爾文男爵威廉·湯姆森|開爾文勳爵]]。
第一类  <math>ber_v(x)</math>,<math>bei_v(x)</math>

第二类  <math>ker_v(x)</math>,<math>kei_v(x)</math>

==第一类开尔文函数==
[[File:KelvinBer.png|thumb|Kelvin Ber(v,z)  function]]
[[File:KelvinBei.png|thumb|Kelvin Bei(v,z)  function]]
<math>ber_v(x)=Re(J_\nu \left (x e^{\frac{3 \pi i}{4}} \right ))</math>

其中<math>Re(f)</math>代表<math>f</math>的实数部分，<math>J_\nu</math>是[[贝塞尔函数#第一类贝塞尔函数|第一类贝塞尔函数]]。

<math>ber_v(x)=Im(J_\nu \left (x e^{\frac{3 \pi i}{4}} \right ))</math>

其中<math>Im(f)</math>代表<math>f</math>的虚数部分。

:<math>\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</math>

:<math>\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</math>

其中n为整数，上式分母的<math>\Gamma</math>是[[Γ函数]]。

==第二类开尔文函数==
[[File:KelvinKer.png|thumb|Kelvin Ker(v,z)  function]]
[[File:KelvinKei.png|thumb|Kelvin Kei(v,z)  function]]

<math>Ker_v(x)=Re(  K_\nu \left (x e^{\frac{\pi i}{4}} \right ) )</math>

<math>Kei_v(x)=Im(  K_\nu \left (x e^{\frac{\pi i}{4}} \right ) )</math>
其中<math>K_\nu</math>是[[贝塞尔函数#修正贝塞尔函数|第二類修正贝塞尔函数]]。

:<math>\begin{align}
\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{align}</math>

:<math>\begin{align}
\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{align}</math>

上式的<math>\psi</math>是[[双伽玛函数]]。
==参考文献==
<references/>
*{{citation|publisher=Cambridge University Press|year=2010|title=NIST Handbook of Mathematical Functions|editor-first=Frank W. J.|editor-last=Olver|editor2-first=Daniel W.|editor2-last=Lozier|editor3-first=Ronald F.|editor3-last=Boisvert|editor4-first=Charles W.|editor4-last=Clark|displayeditors=4|isbn=978-0-521-19225-5|url=http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=9780521140638|mr=2723248|accessdate=2015-01-24|archive-date=2013-07-03|archive-url=https://archive.today/20130703230148/http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=9780521140638|dead-url=no}}

[[Category:特殊超几何函数]]