Struve H function 2D animation Struve L function Struve function H 3D plot Struve L function Struve H function complex plot 司徒盧威函數 （H α x )），滿足下列非齊次貝塞爾方程 ：
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{\displaystyle 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)}}}
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{\displaystyle \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}}
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{\displaystyle \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 [ 編輯 ]
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{\displaystyle \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 [ 編輯 ] 司徒盧威函數滿足下列歸遞關係
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{\displaystyle {\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 [ 編輯 ] 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 978-0-486-61272-0LCCN 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 978-0521192255MR 2723248 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 
