查看“︁司徒卢威函数”︁的源代码

[[File:StruveH2d.gif|thumb|Struve  H function 2D  animation]]
[[File:StruveL2d.gif|thumb|Struve L function]]
[[File:StruveH.gif|thumb|Struve function H  3D  plot]]
[[File:StruveL.gif|thumb|Struve L function]]
[[File:StruveH3dcomplex.png|thumb|Struve  H function complex plot]]

'''司徒卢威函数'''（'''H'''<sub>''α''</sub>(''x'')），满足下列非齐次[[贝塞尔方程]]：

: <math>x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + \left (x^2 - \alpha^2 \right )y = \frac{4\left (\frac{x}{2}\right)^{\alpha+1}}{\sqrt{\pi}\Gamma \left (\alpha+\frac{1}{2} \right )}</math>


:<math> \mathbf{H}_\alpha(x) = \sum_{m=0}^\infty \frac{(-1)^m}{\Gamma \left (m+\frac{3}{2} \right ) \Gamma \left (m+\alpha+\frac{3}{2} \right )} \left({\frac{x}{2}}\right)^{2m+\alpha+1} </math>


变形司徒卢威函数

:<math> \mathbf{L}_{\nu}(z) = \left({\frac{z}{2}}\right)^{\nu+1} \sum_{k=0}^\infty \frac{1}{\Gamma \left (\frac{3}{2}+k \right ) \Gamma \left (\frac{3}{2}+k+\nu \right )} \left(\frac{z}{2}\right)^{2k} </math>

==积分式==
:<math>\mathbf{H}_\alpha(x) = \frac{2\left (\frac{x}{2}\right)^{\alpha}}{\sqrt{\pi}\Gamma \left (\alpha+\frac{1}{2} \right )} \int_{0}^{\frac{\pi}{2}} \sin (x \cos \tau)\sin^{2\alpha}(\tau) d\tau.</math>

==归递式==
司徒卢威函数满足下列归递关系
:<math>\begin{align}
\mathbf{H}_{\alpha -1}(x) + \mathbf{H}_{\alpha+1}(x) &= \frac{2\alpha}{x} \mathbf{H}_\alpha (x) + \frac{\left (\frac{x}{2}\right)^{\alpha}}{\sqrt{\pi}\Gamma \left (\alpha + \frac{3}{2} \right )}, \\
\mathbf{H}_{\alpha -1}(x) - \mathbf{H}_{\alpha+1}(x) &= 2 \frac{d}{dx} \left (\mathbf{H}_\alpha(x) \right) - \frac{ \left( \frac{x}{2} \right)^\alpha}{\sqrt{\pi}\Gamma \left (\alpha + \frac{3}{2} \right )}.
\end{align}</math>




==参考文献==
*{{cite journal|doi=10.1121/1.1564019|author=R.M. Aarts and Augustus J.E.M. Janssen|title=Approximation of the Struve function H1 occurring in impedance calculations|journal= J. Acoust. Soc. Am.|volume= 113 |pages= 2635–2637 |year= 2003| pmid=12765381 | issue=5|bibcode = 2003ASAJ..113.2635A }}
*{{AS ref|12|496}}
*{{springer|id=S/s090700|first=A.B. |last=Ivanov}}
*{{dlmf|id=11|Struve and Related Functions|first=R. B. |last=Paris}}
*{{cite journal|doi=10.1002/andp.18822531319 |first=H. |last=Struve |title=Beitrag zur Theorie der Diffraction an Fernröhren |journal= Ann. Physik Chemie |volume= 17|issue=13 |year=1882 |pages= 1008–1016|bibcode = 1882AnP...253.1008S }}


[[Category:特殊函数]]
[[Category:斯特魯維家族]]