Struve function

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In mathematics, Struve functions $\mathcal{H}_\alpha(x)$ or H_α(x), are solutions y(x) of the non-homogenous Bessel's differential equation:

$x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - \alpha^2)y = \frac{4{(x/2)}^{\alpha+1}}{\sqrt{\pi}\Gamma(\alpha+\frac{1}{2})}$

introduced by H. 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/2H_α(ix).

1 Definitions
- 1.1 Power series expansion
- 1.2 Integral form
2 Asymptotic forms
3 Properties
4 Relation to other functions
5 References
6 External links

[edit] Definitions

Since this is a 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 functions, and the particular solution may be chosen as the corresponding Struve function.

[edit] Power series expansion

Struve functions, denoted as $\mathcal{H}_\alpha(x)$ have the following power series form

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

where $Γ(z)$ is the gamma function.

[edit] Integral form

Another definition of the Struve function, for values of $α$ satisfying $\Re\alpha > -1/2$ , is possible using an integral representation:

$\mathcal{H}_\alpha(x) = \frac{2{(x/2)}^{\alpha}}{\sqrt{\pi}\Gamma(\alpha+\frac{1}{2})} \int_{0}^{\pi/2} \sin (x \sin \tau)\sin^{2\alpha}(\tau) d\tau.$

[edit] Asymptotic forms

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

For large x, one obtains:

$\mathcal{H}_\alpha(x) - Y_\alpha(x) \rightarrow \frac{1}{\sqrt{\pi}\Gamma(\alpha+\frac{1}{2})} {\left(\frac{x}{2}\right)}^{\alpha-1} + O\left({(x/2)}^{\alpha-3}\right)$

where $Y α (x)$ is the Neumann function.

[edit] Properties

The Struve functions satisfy the following recurrence relations:

$\mathcal{H}_{\alpha -1}(x) + \mathcal{H}_{\alpha+1}(x) = \frac{2\alpha}{x} \mathcal{H}_\alpha (x) + \frac{{(x/2)}^\alpha}{\sqrt{\pi}\Gamma(\alpha + \frac{3}{2})}$

$\mathcal{H}_{\alpha -1}(x) - \mathcal{H}_{\alpha+1}(x) = 2\frac{\mathrm{d}\mathcal{H}_\alpha}{\mathrm{d}x} - \frac{{(x/2)}^\alpha}{\sqrt{\pi}\Gamma(\alpha + \frac{3}{2})}.$

[edit] Relation to other functions

Struve functions of integer order can be expressed in terms of Weber functions anE_n and vice versa: if n is a non-negative integer then

$\mathbf{E}_n(z)=\frac{1}{\pi} \sum_{k=0}^{[\frac{n-1}{2}]}\frac{\Gamma(k+1/2)(z/2)^{n-2k-1}}{\Gamma(n-1/2-k)}\mathbf{H}_n$

$\mathbf{E}_{-n}(z)=\frac{(-1)^{n+1}}{\pi}\sum_{k=0}^{[\frac{n-1}{2}]} \frac{\Gamma(n-k-1/2)(z/2)^{-n+2k+1}}{\Gamma(k+3/2)}\mathbf{H}_{-n}.$

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

$\mathbf{H}_{-n-1/2}(z) = (-1)^nJ_{n+1/2}(z)$

where the right hand side is a spherical Bessel function.

Struve functions (of any order) can be expressed in terms of the hypergeometric function ₁F₂ (which is not the Gauss hypergeometric function ₂F₁) :

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

[edit] References

Abramowitz, Milton & Stegun, Irene A., eds. (1965), “Chapter 12”, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, ISBN 0-486-61272-4 .
Ivanov, A.B. (2001), “Struve function”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
Struve, H. (1882), Ann. Physik Chemie 17: 1008–1016