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[[File:Mplwp Struve function05.svg|320px|thumb|Graph of <math>\mathrm{H}_n(x)</math> for <math>n\in [0,1,2,3,4,5]</math>]]
In [[mathematics]], the '''Struve functions''' {{math|'''H'''<sub>''α''</sub>(''x'')}}, are solutions {{math|''y''(''x'')}} of the non-homogeneous [[Bessel's differential equation]]:

: <math>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 )}</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.

And further defined its second-kind version <math>\mathbf{K}_\alpha(x)</math> as <math>\mathbf{K}_\alpha(x)=\mathbf{H}_\alpha(x)-Y_\alpha(x)</math>, where <math>Y_\alpha(x)</math> is the [[Bessel function#Bessel functions of the second kind : Y.CE.B1|Neumann function]].

The '''modified Struve functions''' {{math|'''L'''<sub>''α''</sub>(''x'')}} are equal to {{math|−''ie''<sup>−''iαπ'' / 2</sup>'''H'''<sub>''α''</sub>(''ix'')}} and are solutions {{math|''y''(''x'')}} of the non-homogeneous [[Bessel's differential equation]]:

: [[File: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.svg|alt=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|thumb|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]]<math>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 )}</math>

And further defined its second-kind version <math>\mathbf{M}_\alpha(x)</math> as <math>\mathbf{M}_\alpha(x)=\mathbf{L}_\alpha(x)-I_\alpha(x)</math>, where <math>I_\alpha(x)</math> is the [[Bessel function#Modified Bessel functions: Iα, KαBessel function#Bessel functions of the second kind : Y.CE.B1|modified Bessel function]].

==Definitions==
Since this is a [[Ordinary differential equation#homogeneous|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 function]]s, and the particular solution may be chosen as the corresponding Struve function.

===Power series expansion===
Struve functions, denoted as {{math|'''H'''<sub>''α''</sub>(''z'')}} have the [[power series]] form

:<math> \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},</math>

where {{math|Γ(''z'')}} is the [[gamma function]].

The modified Struve functions, denoted {{math|'''L'''<sub>''α''</sub>(''z'')}}, have the following power series form

:<math> \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}.</math>[[File: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.svg|alt=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|thumb|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 {{mvar|α}} satisfying {{math|Re(''α'') > −&thinsp;{{sfrac|1|2}}}}, is possible expressing in term of the Poisson's integral representation:

<math display="block">\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</math>

<math display="block">\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</math>

<math display="block">\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</math>

<math display="block">\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</math>

==Asymptotic forms==
For small {{mvar|x}}, the power series expansion is given [[#Power series expansion|above]].

For large {{mvar|x}}, one obtains:

:<math>\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),</math>

where {{math|''Y<sub>α</sub>''(''x'')}} 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>\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}</math>

==Relation to other functions==
Struve functions of integer order can be expressed in terms of [[Anger function|Weber function]]s {{math|'''E'''<sub>''n''</sub>}} and vice versa: if {{mvar|n}} is a non-negative integer then

:<math>\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}</math>

Struve functions of  order {{math|''n'' + {{sfrac|1|2}}}} where {{mvar|n}} is an integer can be expressed in terms of elementary functions. In particular if {{mvar|n}} is a non-negative integer then

:<math>\mathbf{H}_{-n-\frac{1}{2}} (z) = (-1)^n J_{n+\frac{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]] {{math|<sub>1</sub>''F''<sub>2</sub>}}:

:<math>\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 ).</math>

==Applications==

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

==References==
{{Reflist}}
*{{cite journal|doi=10.1121/1.1564019 |author=R. M. Aarts and Augustus J. E. M. Janssen |title=Approximation of the Struve function ''H''<sub>1</sub> occurring in impedance calculations |journal= J. Acoust. Soc. Am. |volume= 113 |pages= 2635–2637 |year= 2003 |pmid=12765381 |issue=5 |bibcode = 2003ASAJ..113.2635A }}
*{{cite journal|doi=10.1121/1.4968792 |pmid=28040027 |author=R. M. Aarts and Augustus J. E. M. Janssen |title=Efficient approximation of the Struve functions ''H''<sub>''n''</sub> occurring in the calculation of sound radiation quantities |journal= J. Acoust. Soc. Am. |volume= 140 |pages= 4154–4160 |year= 2016 |issue=6 |bibcode=2016ASAJ..140.4154A|url=https://research.tue.nl/nl/publications/efficient-approximation-of-the-struve-functions-hn-occurring-in-the-calculation-of-sound-radiation-quantaties(c68b8858-9c9d-4ff2-bf39-e888bb638527).html }}
*{{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= Annalen der Physik und Chemie |volume= 17|issue=13 |year=1882 |pages= 1008–1016|bibcode = 1882AnP...253.1008S |url=https://zenodo.org/record/1423790 }}

==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]]