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-In [[mathematics]], '''Struve functions''' <math>\mathbf{H}_\alpha(x)</math>, are  solutions ''y''(''x'') of the non-homogenous [[Bessel's differential equation]]:
+\ո
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-: <math>x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - \alpha^2)y = \frac{4{(x/2)}^{\alpha+1}}{\sqrt{\pi}\Gamma(\alpha+\frac{1}{2})}</math>
-introduced by {{harvs|txt|last=Struve|first=Hermann|authorlink=Hermann Struve|year=1882}}.
-The [[complex number]] α is  the '''order''' of the Struve function, and is often an integer.
-The '''modified Struve functions''' <math>\mathbf{L}_\alpha(x)</math> are equal to
-<math>-i e^{-i\alpha \pi /2} \mathbf{H}_\alpha(ix)</math>.
-==Definitions==
-Since this is a [[Ordinary differential equation#Nonhomogeneous equations|non-homogenous]] equation,
-solutions can be constructed from a single particular solution by adding the solutions of the homogeneous problem.
-In this case, the homogenous solutions are the [[Bessel function]]s, and the particular solution may be chosen as the corresponding Struve function.
-===Power series expansion===
-Struve functions, denoted as <math>\mathbf{H}_\alpha(x)</math> have the following power series form
-:<math> \mathbf{H}_\alpha(x) =
-   \sum_{m=0}^\infty \frac{(-1)^m}{\Gamma(m+\frac{3}{2}) \Gamma(m+\alpha+\frac{3}{2})}
-                                  {\left({\frac{x}{2}}\right)}^{2m+\alpha+1} </math>
-where <math>\Gamma(z)</math> is the [[gamma function]].
-The modified Struve function, denoted as <math>\mathbf{L}_{\nu}(z)</math> have the following power series form
-:<math> \mathbf{L}_{\nu}(z) = {\left({\frac{z}{2}}\right)}^{\nu+1}
-   \sum_{k=0}^\infty \frac{1}{\Gamma(\frac{3}{2}+k) \Gamma(\frac{3}{2}+k+\nu)}
-                                  {\left({\frac{z}{2}}\right)}^{2k} </math>
-===Integral form===
-Another definition of the Struve function, for values of α satisfying <math>\operatorname{Re}\{ \alpha \} > -1/2</math>, is possible using an integral representation:
-:<math>\mathbf{H}_\alpha(x) =
-       \frac{2{(x/2)}^{\alpha}}{\sqrt{\pi}\Gamma(\alpha+\frac{1}{2})}
-       \int_{0}^{\pi/2} \sin (x \cos \tau)\sin^{2\alpha}(\tau) d\tau.</math>
-==Asymptotic forms==
-For small x, the power series expansion is given [[#Power series expansion|above]].
-For large x, one obtains:
-:<math>\mathbf{H}_\alpha(x) - Y_\alpha(x) \rightarrow
-       \frac{1}{\sqrt{\pi}\Gamma(\alpha+\frac{1}{2})} {\left(\frac{x}{2}\right)}^{\alpha-1}
-       + O\left({(x/2)}^{\alpha-3}\right) </math>
-where <math>Y_\alpha(x)</math> is the
-[[Bessel function#Bessel functions of the second kind : Y.CE.B1|Neumann function]].
-==Properties==
-The Struve functions satisfy the following recurrence relations:
-:<math>
-\mathbf{H}_{\alpha -1}(x) + \mathbf{H}_{\alpha+1}(x) =
-   \frac{2\alpha}{x} \mathbf{H}_\alpha (x) + \frac{{(x/2)}^\alpha}{\sqrt{\pi}\Gamma(\alpha + \frac{3}{2})}
-</math>
-:<math>
-\mathbf{H}_{\alpha -1}(x) - \mathbf{H}_{\alpha+1}(x) =
-\frac{\mathrm{d}\mathbf{H}_\alpha}{\mathrm{d}x}  -
-   \frac{{(x/2)}^\alpha}{\sqrt{\pi}\Gamma(\alpha + \frac{3}{2})}.
-</math>
-==Relation to other functions==
-Struve functions of integer order can be expressed in terms of [[Anger function|Weber function]]s '''E'''<sub>''n''</sub> and vice versa: if ''n'' is a non-negative integer then
-:<math>\mathbf{E}_n(z)=\frac{1}{\pi} \sum_{k=0}^{[\frac{n-1}{2}]}\frac{\Gamma(k+1/2)(z/2)^{n-2k-1}}{\Gamma(n-1/2-k)}\mathbf{H}_n </math>
-:<math>\mathbf{E}_{-n}(z)=\frac{(-1)^{n+1}}{\pi}\sum_{k=0}^{[\frac{n-1}{2}]} \frac{\Gamma(n-k-1/2)(z/2)^{-n+2k+1}}{\Gamma(k+3/2)}\mathbf{H}_{-n}. </math>
-Struve functions of  order ''n''+1/2 (''n'' an integer) can be expressed in terms of elementary functions. In particular if ''n'' is a non-negative integer then
-:<math>\mathbf{H}_{-n-1/2}(z) = (-1)^nJ_{n+1/2}(z)</math>
-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]] <sub>1</sub>F<sub>2</sub> (which is '''not''' the  Gauss hypergeometric function <sub>2</sub>F<sub>1</sub>) :
-:<math>\mathbf{H}_{\alpha}(z) = \frac{(z/2)^{\alpha+1/2}}{\sqrt{2\pi}\Gamma(\alpha+3/2)}{}_1F_2(1,3/2,\alpha+3/2,-z^2/4).</math>
-==References==
-*{{cite journal|doi=10.1121/1.1564019|author=R.M. Aarts and Augustus J.E.M. Janssen|title=Approximation of the Struve function H1 occurring in impedance calculations|journal= J. Acoust. Soc. Am. |volume= 113 |pages= 2635–2637 |year= 2003|pmid=12765381|issue=5|bibcode = 2003ASAJ..113.2635A }}
-*{{AS ref|12|496}}
-*{{springer|id=S/s090700|first=A.B. |last=Ivanov}}
-*{{dlmf|id=11|Struve and Related Functions|first=R. B. |last=Paris}}
-*{{cite journal|doi=10.1002/andp.18822531319 |first=H. |last=Struve |title=Beitrag zur Theorie der Diffraction an Fernröhren |journal= Ann. Physik Chemie |volume= 17|issue=13 |year=1882 |pages= 1008–1016|bibcode = 1882AnP...253.1008S }}
-==External links==
-*[http://functions.wolfram.com/Bessel-TypeFunctions/StruveH/introductions/Struves/ Struve functions] at [http://functions.wolfram.com the Wolfram functions site].
-{{DEFAULTSORT:Struve Function}}
-[[Category:Special functions]]
-[[Category:Struve family]]

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