Struve function

[[File:Mplwp Struve function05.svg|320px|thumb|Graph of \mathrm{H}_n(x) for n\in [0,1,2,3,4,5] In mathematics, the Struve functions , are solutions of the non-homogeneous Bessel's differential equation: : 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 )} 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 \mathbf{K}_\alpha(x) as \mathbf{K}_\alpha(x)=\mathbf{H}_\alpha(x)-Y_\alpha(x) , where Y_\alpha(x) is the Neumann function. The modified Struve functions are equal to and are solutions of the non-homogeneous Bessel's differential equation: : 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 )} And further defined its second-kind version \mathbf{M}_\alpha(x) as \mathbf{M}_\alpha(x)=\mathbf{L}_\alpha(x)-I_\alpha(x) , where I_\alpha(x) is the modified Bessel function of the first kind.

[[File:Mplwp Struve function05.svg|320px|thumb|Graph of \mathrm{H}_n(x) for n\in [0,1,2,3,4,5] ]] In mathematics, the Struve functions , are solutions of the non-homogeneous Bessel's differential equation: 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 )} 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 \mathbf{K}_\alpha(x) as \mathbf{K}_\alpha(x)=\mathbf{H}_\alpha(x)-Y_\alpha(x) , where Y_\alpha(x) is the Neumann function. The modified Struve functions are equal to and are solutions of the non-homogeneous Bessel's differential equation: 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 )} And further defined its second-kind version \mathbf{M}_\alpha(x) as \mathbf{M}_\alpha(x)=\mathbf{L}_\alpha(x)-I_\alpha(x) , where I_\alpha(x) is the modified Bessel function of the first kind. ==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 have the power series form \mathbf{H}_\alpha(z) = \sum_{m=0}^\infty \frac{(-1)^m}{\Gamma \left (m+\frac{3}{2} \right ) \Gamma \left (m+\alpha+\frac{3}{2} \right )} \left({\frac{z}{2}}\right)^{2m+\alpha+1}, where is the gamma function. The modified Struve functions, denoted , have the following power series form \mathbf{L}_\alpha(z) = \sum_{m=0}^\infty \frac{1}{\Gamma \left (m+\frac{3}{2} \right ) \Gamma \left (m+\alpha+\frac{3}{2} \right )} \left(\frac{z}{2}\right)^{2m+\alpha+1}. ===Integral form=== Another definition of the Struve function, for values of satisfying , is possible expressing in term of the Poisson's integral representation: \mathbf{H}_\alpha(x)=\frac{2\left(\frac{x}{2}\right)^\alpha}{\sqrt\pi\Gamma\left(\alpha+\frac{1}{2}\right)}\int_0^1(1-t^2)^{\alpha-\frac{1}{2}}\sin xt~dt=\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=\frac{2\left(\frac{x}{2}\right)^\alpha}{\sqrt\pi\Gamma\left(\alpha+\frac{1}{2}\right)}\int_0^\frac{\pi}{2}\sin(x\sin\tau)\cos^{2\alpha}\tau~d\tau \mathbf{K}_\alpha(x)=\frac{2\left(\frac{x}{2}\right)^\alpha}{\sqrt\pi\Gamma\left(\alpha+\frac{1}{2}\right)}\int_0^\infty(1+t^2)^{\alpha-\frac{1}{2}}e^{-xt}~dt=\frac{2\left(\frac{x}{2}\right)^\alpha}{\sqrt\pi\Gamma\left(\alpha+\frac{1}{2}\right)}\int_0^\infty e^{-x\sinh\tau}\cosh^{2\alpha}\tau~d\tau \mathbf{L}_\alpha(x)=\frac{2\left(\frac{x}{2}\right)^\alpha}{\sqrt\pi\Gamma\left(\alpha+\frac{1}{2}\right)}\int_0^1(1-t^2)^{\alpha-\frac{1}{2}}\sinh xt~dt=\frac{2\left(\frac{x}{2}\right)^\alpha}{\sqrt\pi\Gamma\left(\alpha+\frac{1}{2}\right)}\int_0^\frac{\pi}{2}\sinh(x\cos\tau)\sin^{2\alpha}\tau~d\tau=\frac{2\left(\frac{x}{2}\right)^\alpha}{\sqrt\pi\Gamma\left(\alpha+\frac{1}{2}\right)}\int_0^\frac{\pi}{2}\sinh(x\sin\tau)\cos^{2\alpha}\tau~d\tau \mathbf{M}_\alpha(x)=-\frac{2\left(\frac{x}{2}\right)^\alpha}{\sqrt\pi\Gamma\left(\alpha+\frac{1}{2}\right)}\int_0^1(1-t^2)^{\alpha-\frac{1}{2}}e^{-xt}~dt=-\frac{2\left(\frac{x}{2}\right)^\alpha}{\sqrt\pi\Gamma\left(\alpha+\frac{1}{2}\right)}\int_0^\frac{\pi}{2}e^{-x\cos\tau}\sin^{2\alpha}\tau~d\tau=-\frac{2\left(\frac{x}{2}\right)^\alpha}{\sqrt\pi\Gamma\left(\alpha+\frac{1}{2}\right)}\int_0^\frac{\pi}{2}e^{-x\sin\tau}\cos^{2\alpha}\tau~d\tau ==Asymptotic forms== For small , the power series expansion is given above. For large , one obtains: \mathbf{H}_\alpha(x) - Y_\alpha(x) = \frac{\left(\frac{x}{2}\right)^{\alpha-1}}{\sqrt{\pi} \Gamma \left (\alpha+\frac{1}{2} \right )} + O\left(\left (\tfrac{x}{2}\right)^{\alpha-3}\right), where is the Neumann function. ==Properties== The Struve functions satisfy the following recurrence relations: \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} ==Relation to other functions== Struve functions of integer order can be expressed in terms of Weber functions and vice versa: if is a non-negative integer then \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 \mathbf{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 : \mathbf{H}_{\alpha}(z) = \frac{z^{\alpha+1}}{2^{\alpha}\sqrt{\pi} \Gamma \left (\alpha+\tfrac{3}{2} \right )} {}_1F_2 \left (1;\tfrac{3}{2}, \alpha+\tfrac{3}{2};-\tfrac{z^2}{4} \right ). ==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== ==External links== Struve functions at the Wolfram functions site.

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