Struve function

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.

Struve functions, denoted as {{math|H_α(z)}} 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 {{math|Γ(z)}} is the gamma function.

The modified Struve functions, denoted {{math|L_α(z)}}, have the following power series form

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

Another definition of the Struve function, for values of {{mvar|α}} satisfying {{math|Re(α) > − {{sfrac|1|2}}}}, is possible expressing in term of the Poisson's integral representation:

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

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

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

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

For small {{mvar|x}}, the power series expansion is given above.

For large {{mvar|x}}, 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 {{math|Y_α(x)}} is the Neumann function.

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}

Struve functions of integer order can be expressed in terms of Weber functions {{math|E_n}} and vice versa: if {{mvar|n}} 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 {{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

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

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 {{math|₁F₂}}:

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

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

{{Reflist}}

• {{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 }}
• {{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_n 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 }}

• [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