司徒盧威函數

司徒盧威函數（H_α(x)），滿足下列非齊次貝塞爾方程：

Struve H function 2D animation

Struve L function

Struve function H 3D plot

Struve L function

Struve H function complex plot

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

$\mathbf {H} _{\alpha }(x)=\sum _{m=0}^{\infty }{\frac {(-1)^{m}}{\Gamma \left(m+{\frac {3}{2}}\right)\Gamma \left(m+\alpha +{\frac {3}{2}}\right)}}\left({\frac {x}{2}}\right)^{2m+\alpha +1}$

變形司徒盧威函數

$\mathbf {L} _{\nu }(z)=\left({\frac {z}{2}}\right)^{\nu +1}\sum _{k=0}^{\infty }{\frac {1}{\Gamma \left({\frac {3}{2}}+k\right)\Gamma \left({\frac {3}{2}}+k+\nu \right)}}\left({\frac {z}{2}}\right)^{2k}$

積分式

$\mathbf {H} _{\alpha }(x)={\frac {2\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}\int _{0}^{\frac {\pi }{2}}\sin(x\cos \tau )\sin ^{2\alpha }(\tau )d\tau .$

歸遞式

司徒盧威函數滿足下列歸遞關係

${\begin{aligned}\mathbf {H} _{\alpha -1}(x)+\mathbf {H} _{\alpha +1}(x)&={\frac {2\alpha }{x}}\mathbf {H} _{\alpha }(x)+{\frac {\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {3}{2}}\right)}},\\\mathbf {H} _{\alpha -1}(x)-\mathbf {H} _{\alpha +1}(x)&=2{\frac {d}{dx}}\left(\mathbf {H} _{\alpha }(x)\right)-{\frac {\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {3}{2}}\right)}}.\end{aligned}}$

參考文獻

• R.M. Aarts and Augustus J.E.M. Janssen. Approximation of the Struve function H1 occurring in impedance calculations. J. Acoust. Soc. Am. 2003, 113 (5): 2635–2637. Bibcode:2003ASAJ..113.2635A. PMID 12765381. doi:10.1121/1.1564019. 
• Abramowitz, Milton; Stegun, Irene Ann (編). Chapter 12. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series 55 Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first. Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. 1983: 496. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. . 
• Ivanov, A.B., S/s090700, Hazewinkel, Michiel (編), 数学百科全书, Springer, 2001, ISBN 978-1-55608-010-4 
• Paris, R. B., 司徒卢威函数, Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (編), NIST Handbook of Mathematical Functions, Cambridge University Press, 2010, ISBN 978-0521192255, MR2723248 
• Struve, H. Beitrag zur Theorie der Diffraction an Fernröhren. Ann. Physik Chemie. 1882, 17 (13): 1008–1016. Bibcode:1882AnP...253.1008S. doi:10.1002/andp.18822531319.