Definition
In mathematics, a Struve function $ \mathbf{H}_
u(x) $ is a particular solution of the inhomogeneous Bessel differential equation
$$ x^{2}y''+xy'+\left(x^{2}- u^{2}\right)y=\frac{4,(x/2)^{ u+1}}{\sqrt{\pi},\Gamma!\left( u+\tfrac12\right)}, $$
where $ u $ is a real or complex order and $ \Gamma $ denotes the gamma function. The function $ \mathbf{H}_ u(x) $ is named the Struve function of order $ u $.
Overview
Struve functions complement the family of Bessel functions. While Bessel functions $ J_
u(x) $ and $ Y_
u(x) $ solve the homogeneous version of the above equation, Struve functions solve the same equation with a non‑zero right‑hand side, making them useful in various physical problems involving wave propagation, heat conduction, and electromagnetic theory where source terms appear.
For integer orders $ n\ge 0 $ the notation $ \mathbf{H}_n(x) $ is common; for half‑integer and non‑integer orders the same definition applies via analytic continuation.
Etymology / Origin
The functions are named after the German astronomer and mathematician Hermann Struve (1854–1920), who investigated them in the context of solutions to Bessel’s equation with an inhomogeneous term. The name first appeared in the mathematical literature in the early 20th century, notably in Struve’s own papers and subsequent treatises on special functions.
Characteristics
| Property | Description |
|---|---|
| Series representation | For $ |
| u > -\tfrac12 $ and all $ x $, $$ |
|
| \mathbf{H}_ | |
| u(x)=\sum_{k=0}^{\infty}\frac{(-1)^k}{\Gamma!\left(k+\tfrac32\right)\Gamma!\left(k+ | |
| u+\tfrac32\right)}\left(\frac{x}{2}\right)^{2k+ | |
| u+1}. | |
| $$ | |
| Integral representation | For $ \Re( |
| u)>-1/2 $, $$ |
|
| \mathbf{H}_ | |
| u(x)=\frac{2\left(\frac{x}{2}\right)^{ | |
| u}}{\sqrt{\pi},\Gamma!\left( | |
| u+\tfrac12\right)}\int_{0}^{\pi/2}\sin!\bigl(x\cos\theta\bigr),\sin^{2 | |
| u}\theta;d\theta. | |
| $$ | |
| Relation to Bessel functions | The Struve function can be expressed via Lommel functions or as a linear combination of Bessel functions and an integral term. For integer $ n $: $$ |
| \mathbf{H}n(x)=J_n(x)-\frac{2}{\pi}\int{0}^{x} \frac{J_n(t)}{t},dt. | |
| $$ | |
| Asymptotic behavior | As $ x\to\infty $: $$ |
| \mathbf{H}_ | |
| u(x)\sim 1-\frac{ | |
| u( | |
| u-1)}{2x^{2}}+\mathcal{O}!\left(x^{-4}\right), | |
| $$ while for small arguments $ x\to0 $: $$ |
|
| \mathbf{H}_ | |
| u(x)\sim \frac{2^{ | |
| u-1}}{\sqrt{\pi},\Gamma!\left( | |
| u+\tfrac32\right)},x^{ | |
| u+1}+ \mathcal{O}!\left(x^{ | |
| u+3}\right). | |
| $$ | |
| Differential equation | Satisfies the inhomogeneous Bessel equation stated in the definition. |
| Special cases | $ \mathbf{H}{-1/2}(x)=\sqrt{\tfrac{2}{\pi x}}\left(1-\cos x\right) $, $ \mathbf{H}{1/2}(x)=\sqrt{\tfrac{2}{\pi x}}\sin x $. |
| Numerical evaluation | Implemented in major mathematical libraries (e.g., SciPy’s scipy.special.struve, MATLAB’s struve, and the GNU Scientific Library). |
Related Topics
- Bessel functions $ J_ u(x) $ and $ Y_ u(x) $ – homogeneous solutions of the same differential equation.
- Lommel functions – another class of solutions to inhomogeneous Bessel equations.
- Neumann functions – also known as Bessel functions of the second kind.
- Special functions – broader category encompassing hypergeometric, Airy, and Legendre functions.
- Hermann Struve – biographical information on the mathematician after whom the functions are named.
References
- Watson, G. N. A Treatise on the Theory of Bessel Functions. Cambridge University Press, 1944.
- Abramowitz, M., and Stegun, I. A. (eds.). Handbook of Mathematical Functions. Dover Publications, 1972.
- Olver, F. W. J., et al. NIST Handbook of Mathematical Functions. Cambridge University Press, 2010.
- DLMF, “Struve Functions”, https://dlmf.nist.gov/11.2.