司徒卢威函数

Struve H function 2D animation

Struve L function

File:StruveH.gif

Struve function H 3D plot

Struve L function

File:StruveH3dcomplex.png

Struve H function complex plot

司徒卢威函数（H_α(x)），满足下列非齐次贝塞尔方程：

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

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

变形司徒卢威函数

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

积分式

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

归递式

司徒卢威函数满足下列归递关系

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

参考文献

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