Struve function

 ( Special Functions ) 

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

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

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

And further defined its second-kind version $\backslash mathbf\{K\}\_\backslash alpha(x)$ as $\backslash mathbf\{K\}\_\backslash alpha(x)=\backslash mathbf\{H\}\_\backslash alpha(x)-Y\_\backslash alpha(x)$, where $Y\_\backslash alpha(x)$ is the .

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

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

And further defined its second-kind version $\backslash mathbf\{M\}\_\backslash alpha(x)$ as $\backslash mathbf\{M\}\_\backslash alpha(x)=\backslash mathbf\{L\}\_\backslash alpha(x)-I\_\backslash alpha(x)$, where $I\_\backslash alpha(x)$ is the .

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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 , and the particular solution may be chosen as the corresponding Struve function.

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Power series expansion Struve functions, denoted as have the power series form

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

where is the gamma function.

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

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

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

$$\backslash mathbf\{H\}\_\backslash alpha(x)=\backslash frac\{2\backslash left(\backslash frac\{x\}\{2\}\backslash right)^\backslash alpha\}\{\backslash sqrt\backslash pi\backslash Gamma\backslash left(\backslash alpha+\backslash frac\{1\}\{2\}\backslash right)\}\backslash int\_0^1(1-t^2)^\{\backslash alpha-\backslash frac\{1\}\{2\}\}\backslash sin\; xt~dt=\backslash frac\{2\backslash left(\backslash frac\{x\}\{2\}\backslash right)^\backslash alpha\}\{\backslash sqrt\backslash pi\backslash Gamma\backslash left(\backslash alpha+\backslash frac\{1\}\{2\}\backslash right)\}\backslash int\_0^\backslash frac\{\backslash pi\}\{2\}\backslash sin(x\backslash cos\backslash tau)\backslash sin^\{2\backslash alpha\}\backslash tau~d\backslash tau=\backslash frac\{2\backslash left(\backslash frac\{x\}\{2\}\backslash right)^\backslash alpha\}\{\backslash sqrt\backslash pi\backslash Gamma\backslash left(\backslash alpha+\backslash frac\{1\}\{2\}\backslash right)\}\backslash int\_0^\backslash frac\{\backslash pi\}\{2\}\backslash sin(x\backslash sin\backslash tau)\backslash cos^\{2\backslash alpha\}\backslash tau~d\backslash tau$$

$$\backslash mathbf\{K\}\_\backslash alpha(x)=\backslash frac\{2\backslash left(\backslash frac\{x\}\{2\}\backslash right)^\backslash alpha\}\{\backslash sqrt\backslash pi\backslash Gamma\backslash left(\backslash alpha+\backslash frac\{1\}\{2\}\backslash right)\}\backslash int\_0^\backslash infty(1+t^2)^\{\backslash alpha-\backslash frac\{1\}\{2\}\}e^\{-xt\}~dt=\backslash frac\{2\backslash left(\backslash frac\{x\}\{2\}\backslash right)^\backslash alpha\}\{\backslash sqrt\backslash pi\backslash Gamma\backslash left(\backslash alpha+\backslash frac\{1\}\{2\}\backslash right)\}\backslash int\_0^\backslash infty\; e^\{-x\backslash sinh\backslash tau\}\backslash cosh^\{2\backslash alpha\}\backslash tau~d\backslash tau$$

$$\backslash mathbf\{L\}\_\backslash alpha(x)=\backslash frac\{2\backslash left(\backslash frac\{x\}\{2\}\backslash right)^\backslash alpha\}\{\backslash sqrt\backslash pi\backslash Gamma\backslash left(\backslash alpha+\backslash frac\{1\}\{2\}\backslash right)\}\backslash int\_0^1(1-t^2)^\{\backslash alpha-\backslash frac\{1\}\{2\}\}\backslash sinh\; xt~dt=\backslash frac\{2\backslash left(\backslash frac\{x\}\{2\}\backslash right)^\backslash alpha\}\{\backslash sqrt\backslash pi\backslash Gamma\backslash left(\backslash alpha+\backslash frac\{1\}\{2\}\backslash right)\}\backslash int\_0^\backslash frac\{\backslash pi\}\{2\}\backslash sinh(x\backslash cos\backslash tau)\backslash sin^\{2\backslash alpha\}\backslash tau~d\backslash tau=\backslash frac\{2\backslash left(\backslash frac\{x\}\{2\}\backslash right)^\backslash alpha\}\{\backslash sqrt\backslash pi\backslash Gamma\backslash left(\backslash alpha+\backslash frac\{1\}\{2\}\backslash right)\}\backslash int\_0^\backslash frac\{\backslash pi\}\{2\}\backslash sinh(x\backslash sin\backslash tau)\backslash cos^\{2\backslash alpha\}\backslash tau~d\backslash tau$$

$$\backslash mathbf\{M\}\_\backslash alpha(x)=-\backslash frac\{2\backslash left(\backslash frac\{x\}\{2\}\backslash right)^\backslash alpha\}\{\backslash sqrt\backslash pi\backslash Gamma\backslash left(\backslash alpha+\backslash frac\{1\}\{2\}\backslash right)\}\backslash int\_0^1(1-t^2)^\{\backslash alpha-\backslash frac\{1\}\{2\}\}e^\{-xt\}~dt=-\backslash frac\{2\backslash left(\backslash frac\{x\}\{2\}\backslash right)^\backslash alpha\}\{\backslash sqrt\backslash pi\backslash Gamma\backslash left(\backslash alpha+\backslash frac\{1\}\{2\}\backslash right)\}\backslash int\_0^\backslash frac\{\backslash pi\}\{2\}e^\{-x\backslash cos\backslash tau\}\backslash sin^\{2\backslash alpha\}\backslash tau~d\backslash tau=-\backslash frac\{2\backslash left(\backslash frac\{x\}\{2\}\backslash right)^\backslash alpha\}\{\backslash sqrt\backslash pi\backslash Gamma\backslash left(\backslash alpha+\backslash frac\{1\}\{2\}\backslash right)\}\backslash int\_0^\backslash frac\{\backslash pi\}\{2\}e^\{-x\backslash sin\backslash tau\}\backslash cos^\{2\backslash alpha\}\backslash tau~d\backslash tau$$

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Asymptotic forms For small , the power series expansion is given above.

For large , one obtains:

$\backslash mathbf\{H\}\_\backslash alpha(x)\; -\; Y\_\backslash alpha(x)\; =\; \backslash frac\{\backslash left(\backslash frac\{x\}\{2\}\backslash right)^\{\backslash alpha-1\}\}\{\backslash sqrt\{\backslash pi\}\; \backslash Gamma\; \backslash left\; (\backslash alpha+\backslash frac\{1\}\{2\}\; \backslash right\; )\}\; +\; O\backslash left(\backslash left\; (\backslash tfrac\{x\}\{2\}\backslash right)^\{\backslash alpha-3\}\backslash right),$

where is the .

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Properties The Struve functions satisfy the following recurrence relations:

$\backslash begin\{align\}$

\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}

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Relation to other functions Struve functions of integer order can be expressed in terms of Anger function and vice versa: if is a non-negative integer then

$\backslash begin\{align\}$

\mathbf{E}_n(z) &= \frac{1}{\pi} \sum_{k=0}^{\left \lfloor \frac{n-1}{2} \right \rfloor} \frac{\Gamma \left (k+ \frac{1}{2} \right) \left (\frac{z}{2} \right )^{n-2k-1}}{\Gamma \left (n- k + \frac{1}{2}\right )} -\mathbf{H}_n(z),\\ \mathbf{E}_{-n}(z) &= \frac{(-1)^{n+1}}{\pi}\sum_{k=0}^{\left \lceil \frac{n-3}{2} \right \rceil} \frac{\Gamma(n-k-\frac{1}{2}) \left (\frac{z}{2} \right )^{-n+2k+1}}{\Gamma \left (k+ \frac{3}{2} \right)}-\mathbf{H}_{-n}(z). \end{align}

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

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

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 :

$\backslash mathbf\{H\}\_\{\backslash alpha\}(z)\; =\; \backslash frac\{z^\{\backslash alpha+1\}\}\{2^\{\backslash alpha\}\backslash sqrt\{\backslash pi\}\; \backslash Gamma\; \backslash left\; (\backslash alpha+\backslash tfrac\{3\}\{2\}\; \backslash right\; )\}\; \{\}\_1F\_2\; \backslash left\; (1;\backslash tfrac\{3\}\{2\},\; \backslash alpha+\backslash tfrac\{3\}\{2\};-\backslash tfrac\{z^2\}\{4\}\; \backslash right\; ).$

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Applications The Struve and Weber functions were shown to have an application to beamforming in.,K. Buchanan, C. Flores, S. Wheeland, J. Jensen, D. Grayson and G. Huff, "Transmit beamforming for radar applications using circularly tapered random arrays," 2017 IEEE Radar Conference (RadarConf), 2017, pp. 0112-0117, doi: 10.1109/RADAR.2017.7944181 and in describing the effect of confining interface on Brownian motion of colloidal particles at low Reynolds numbers.B. U. Felderhof, "Effect of the wall on the velocity autocorrelation function and long-time tail of Brownian motion." The Journal of Physical Chemistry B 109.45, 2005, pp. 21406-21412

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External links

• Struve functions at the Wolfram functions site.

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