Struve function

Graph of ${\mathrm{H}}_n(x)$ for $n\in [0,1,2,3,4,5]$

In mathematics, the Struve functions Template:Math, are solutions Template:Math of the non-homogeneous Bessel's differential equation:

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

introduced by Template:Harvs. The complex number α is the order of the Struve function, and is often an integer.

And further defined its second-kind version ${𝐊}_{\alpha }(x)$ as ${𝐊}_{\alpha }(x)={𝐇}_{\alpha }(x)-Y_{\alpha }(x)$.

The modified Struve functions Template:Math are equal to Template:Math and are solutions Template:Math of the non-homogeneous Bessel's differential equation:

Plot of the Struve function H n(z) with n=2 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D

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

And further defined its second-kind version ${𝐌}_{\alpha }(x)$ as ${𝐌}_{\alpha }(x)={𝐋}_{\alpha }(x)-I_{\alpha }(x)$.

Definitions

Since this is a non-homogeneous equation, solutions can be constructed from a single particular solution by adding the solutions of the homogeneous problem. In this case, the homogeneous solutions are the Bessel functions, and the particular solution may be chosen as the corresponding Struve function.

Power series expansion

Struve functions, denoted as Template:Math have the power series form

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

where Template:Math is the gamma function.

The modified Struve functions, denoted Template:Math, have the following power series form

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

Plot of the modified Struve function L n(z) with n=2 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D

Integral form

Another definition of the Struve function, for values of Template:Mvar satisfying Template:Math, is possible expressing in term of the Poisson's integral representation:

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

$${𝐊}_{\alpha }(x)=\frac{2{\left(\frac{x}{2}\right)}^{\alpha }}{\sqrt{\pi }\mathrm{\Gamma }(\alpha +\frac{1}{2})}\underset{0}{\overset{\mathrm{\infty }}{\int }}(1+t^2{)}^{\alpha -\frac{1}{2}}{e}^{-xt}dt=\frac{2{\left(\frac{x}{2}\right)}^{\alpha }}{\sqrt{\pi }\mathrm{\Gamma }(\alpha +\frac{1}{2})}\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-x\sinh \tau }{\cosh }^{2\alpha }\tau d\tau$$

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

$${𝐌}_{\alpha }(x)=-\frac{2{\left(\frac{x}{2}\right)}^{\alpha }}{\sqrt{\pi }\mathrm{\Gamma }(\alpha +\frac{1}{2})}\underset{0}{\overset{1}{\int }}(1-t^2{)}^{\alpha -\frac{1}{2}}{e}^{-xt}dt=-\frac{2{\left(\frac{x}{2}\right)}^{\alpha }}{\sqrt{\pi }\mathrm{\Gamma }(\alpha +\frac{1}{2})}\underset{0}{\overset{\frac{\pi }{2}}{\int }}{e}^{-x\cos \tau }{\sin }^{2\alpha }\tau d\tau =-\frac{2{\left(\frac{x}{2}\right)}^{\alpha }}{\sqrt{\pi }\mathrm{\Gamma }(\alpha +\frac{1}{2})}\underset{0}{\overset{\frac{\pi }{2}}{\int }}{e}^{-x\sin \tau }{\cos }^{2\alpha }\tau d\tau$$

Asymptotic forms

For small Template:Mvar, the power series expansion is given above.

For large Template:Mvar, one obtains:

${𝐇}_{\alpha }(x)-Y_{\alpha }(x)=\frac{{\left(\frac{x}{2}\right)}^{\alpha -1}}{\sqrt{\pi }\mathrm{\Gamma }(\alpha +\frac{1}{2})}+O\left({\left(\frac{x}{2}\right)}^{\alpha -3}\right),$

where Template:Math is the Neumann function.

Properties

The Struve functions satisfy the following recurrence relations:

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

Relation to other functions

Struve functions of integer order can be expressed in terms of Weber functions Template:Math and vice versa: if Template:Mvar is a non-negative integer then

$\begin{array}{cc}{𝐄}_n(z)& =\frac{1}{\pi }\sum _{k=0}^{\lfloor \frac{n-1}{2}\rfloor }\frac{\mathrm{\Gamma }(k+\frac{1}{2}){\left(\frac{z}{2}\right)}^{n-2k-1}}{\mathrm{\Gamma }(n-k+\frac{1}{2})}-{𝐇}_n(z),\\ {𝐄}_{-n}(z)& =\frac{(-1{)}^{n+1}}{\pi }\sum _{k=0}^{\lceil \frac{n-3}{2}\rceil }\frac{\mathrm{\Gamma }(n-k-\frac{1}{2}){\left(\frac{z}{2}\right)}^{-n+2k+1}}{\mathrm{\Gamma }(k+\frac{3}{2})}-{𝐇}_{-n}(z).\end{array}$

Struve functions of order Template:Math where Template:Mvar is an integer can be expressed in terms of elementary functions. In particular if Template:Mvar is a non-negative integer then

${𝐇}_{-n-\frac{1}{2}}(z)=(-1{)}^n{J}_{n+\frac{1}{2}}(z),$

where the right hand side is a spherical Bessel function.

Struve functions (of any order) can be expressed in terms of the generalized hypergeometric function Template:Math:

${𝐇}_{\alpha }(z)=\frac{z^{\alpha +1}}{2^{\alpha }\sqrt{\pi }\mathrm{\Gamma }(\alpha +\frac{3}{2})}{}_1{F}_2(1;\frac{3}{2},\alpha +\frac{3}{2};-\frac{z^2}{4}).$

Applications

The Struve and Weber functions were shown to have an application to beamforming in., and in describing the effect of confining interface on Brownian motion of colloidal particles at low Reynolds numbers.

References

• Template:Cite journal
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• Template:AS ref
• Template:Springer
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External links

• Struve functions at the Wolfram functions site.