Struve function

In mathematics, Struve functions $H α (x)$ , are solutions $y (x)$ 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 Hermann Struve (1882). The complex number α is the order of the Struve function, and is often an integer. The modified Struve functions $L α (x)$ are equal to $- ie - iαπ / 2 H α (ix)$ .

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 $H α (x)$ have the following power series form

\mathbf{H}_\alpha(x) = \sum_{m=0}^\infty \frac{(-1)^m}{\Gamma \left (m+\frac{3}{2} \right ) \Gamma \left (m+\alpha+\frac{3}{2} \right )} \left({\frac{x}{2}}\right)^{2m+\alpha+1}

where $Γ(z)$ is the gamma function.

The modified Struve function, denoted as $L ν (z)$ have the following power series form

\mathbf{L}_{\nu}(z) = \left({\frac{z}{2}}\right)^{\nu+1} \sum_{k=0}^\infty \frac{1}{\Gamma \left (\frac{3}{2}+k \right ) \Gamma \left (\frac{3}{2}+k+\nu \right )} \left(\frac{z}{2}\right)^{2k}

Integral form

Another definition of the Struve function, for values of $α$ satisfying $Re(α) > - 1 / 2$ , is possible using an 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}^{\frac{\pi}{2}} \sin (x \cos \tau)\sin^{2\alpha}(\tau) d\tau.

Asymptotic forms

For small $x$ , the power series expansion is given above.

For large $x$ , one obtains:

\mathbf{H}_\alpha(x) - Y_\alpha(x) \to \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 $Y α (x)$ 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 $E n$ and vice versa: if $n$ 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 \\ \mathbf{E}_{-n}(z) &= \frac{(-1)^{n+1}}{\pi}\sum_{k=0}^{\left \lfloor \frac{n-1}{2} \right \rfloor} \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}. \end{align}

Struve functions of order $n + 1 / 2$ where $n$ is an integer can be expressed in terms of elementary functions. In particular if $n$ 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 $1 F 2$ (which is not the Gauss hypergeometric function $2 F 1$ ):

\mathbf{H}_{\alpha}(z) = \frac{\left (\frac{z}{2} \right )^{\alpha+\frac{1}{2}}}{\sqrt{2\pi} \Gamma \left (\alpha+\tfrac{3}{2} \right )} {}_1F_2 \left (1,\tfrac{3}{2}, \alpha+\tfrac{3}{2},-\tfrac{z^2}{4} \right ).

References

R.M. Aarts and Augustus J.E.M. Janssen (2003). "Approximation of the Struve function H1 occurring in impedance calculations". J. Acoust. Soc. Am. 113 (5): 2635–2637. Bibcode:2003ASAJ..113.2635A. doi:10.1121/1.1564019. PMID 12765381.
Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 12", Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, p. 496, ISBN 978-0486612720, MR 0167642 .
Ivanov, A.B. (2001), "S/s090700", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Paris, R. B. (2010), "Struve function", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248
Struve, H. (1882). "Beitrag zur Theorie der Diffraction an Fernröhren". Ann. Physik Chemie 17 (13): 1008–1016. Bibcode:1882AnP...253.1008S. doi:10.1002/andp.18822531319.

External links

Struve functions at the Wolfram functions site.