Struve function

2020 Mathematics Subject Classification: Primary: 34-XX [MSN][ZBL]

The function \[ {\bf H}_\nu (x) = \frac{2\left(\textstyle{\frac{x}{2}}\right)^\nu}{\Gamma \left(\nu + \textstyle{\frac{1}{2}}\right) \Gamma \left(\textstyle{\frac{1}{2}}\right)} \int_0^{\frac{\pi}{2}} \, \sin (x\, \cos \theta)\, \sin^{2\nu} \theta\, d\theta\, , \] where $\nu$ is a complex parameter with ${\rm Re}\, \nu > \frac{1}{2}$ and $x$ a complex variable. It was introduced by H. Struve in [S] and it is therefore sometimes denoted by $S_\nu$.

The Struve function satisfies the inhomogeneous Bessel equation: \[ x^2 y'' + x y' + (x^2 - \nu^2) y = \frac{4 \left(\textstyle{\frac{x}{2}}\right)^{\nu+1}}{\Gamma \left(\nu + \textstyle{\frac{1}{2}}\right) \Gamma \left(\textstyle{\frac{1}{2}}\right)} \] (see 10.4 in [Wa]).

The Struve function has the expansion \begin{equation}\label{e:expansion} {\bf H}_\nu (x) = \left(\frac{x}{2}\right)^{\nu +1} \sum_{k=0}^\infty (-1)^k \frac{\left(\textstyle{\frac{x}{2}}\right)^{2k}}{\Gamma \left(k+ \textstyle{\frac{3}{2}}\right) \Gamma \left(\nu + k + \textstyle{\frac{3}{2}}\right)}\, . \end{equation} The Struve functions of integral order $n$ is related to the Weber function ${\bf E}_n$ by the following relation: \[ {\bf E}_n (x) = \sum_{m=1}^n \frac{e^{\frac{1}{2} (m-1) \pi i} \left(\textstyle{\frac{x}{2}}\right)^{n-k}}{\Gamma \left(1-\textstyle{\frac{m}{2}}\right) \Gamma \left(n+1 - \textstyle{\frac{m}{2}}\right)} - {\bf H}_n (x) \quad \mbox{for }\; n\geq 0 \] \[ {\bf E}_{-n} (x) = \frac{(-1)^{n+1}}{\pi} \sum_{0 \leq m < \frac{n}{2}} \frac{\Gamma \left( n - m - \textstyle{\frac{1}{2}}\right) \left(\frac{x}{2}\right)^{-n+2m+1}}{\Gamma \left(m + \textstyle{\frac{3}{2}}\right)} - {\bf H}_{-n} (x)\, \quad \mbox{for }\; n > 0\, , \] (see 10.44 of [Wa]).

The Struve function of order $n + \frac{1}{2}$ with integer $n$ can be expressed in terms of elementary functions. For instance \begin{align*} {\bf H}_{1/2} (x) &= \left(\frac{2}{\pi x}\right)^{\frac{1}{2}} (1-\cos x)\\ {\bf H}_{3/2} (x) &= \left(\frac{x}{2\pi}\right)^{\frac{1}{2}} \left(1+\frac{2}{x^2}\right) - \left(\frac{2}{\pi x}\right)^{\frac{1}{2}} \left(\sin x + \frac{\cos x}{x}\right) \end{align*} (cf. 10.42 of [Wa]).

For $|{\rm arg}\, x| < \pi$ and $|x|$ large we have the asymptotic expansion \[ {\bf H}_\nu (x) = Y_\nu (x) + \frac{1}{\pi} \sum_{m=0}^{k-1} \frac{\Gamma \left(m + \textstyle{\frac{1}{2}}\right)}{\Gamma \left(\nu + \textstyle{\frac{1}{2}} - m\right) \left(\textstyle{\frac{x}{2}}\right)^{2m - \nu +1}} + O \left(|x|^{\nu - 2k -1}\right)\, , \] where $Y_\nu$ is the Neumann function.

The modified Struve function is given by \[ {\bf L}_\nu (x) = \left\{\begin{array}{ll} &e^{-\frac{1}{2} \nu \pi i} {\bf H}_\nu ( ix) \quad &\mbox{if } -\pi < {\rm arg}\, z \leq \frac{\pi}{2}\\ &e^{\frac{3}{2} \nu \pi i} {\bf H}_\nu ( -ix) \quad &\mbox{if } \frac{\pi}{2} < {\rm arg}\, z \leq \pi \end{array}\right. \] and thus bears the same relation to the Struwe function ${\bf H}_\nu (x)$ as the modified Bessel function $I_\nu$ bears to the Bessel function $J_\nu$ (see Cylinder functions).

The expansion \eqref{e:expansion} translates into a corresponding expansion for the modified Struve function. We have moreover the interesting relation \[ {\bf L}_\nu (x) = I_{-\nu} (x) - \frac{2 \left(\textstyle{\frac{x}{2}}\right)^\nu}{\Gamma \left( \nu + \textstyle{\frac{1}{2}}\right) \Gamma \left(\textstyle{\frac{1}{2}}\right)} \int_0^\infty \sin (xu)\, (1+ u^2)^{-\nu - \frac{1}{2}}\, du \] which leads to the asymptotic expansion \[ {\bf L}_\nu (x) = I_{-\nu} (x) - \frac{\left(\textstyle{\frac{x}{2}}\right)^{\nu-1}}{\sqrt{\pi}\,\Gamma \left(\nu + \textstyle{\frac{1}{2}}\right)} \] for $|x|$ large.

The Struve function can be expressed in terms of a hypergeometric function of type $_1 F_2$, cf. [AS], formula (7.5).

References

[AS]	M. Abramowitz, I.A. Stegun, "Handbook of mathematical functions" , Dover, reprint (1970)
[BE]	H. Bateman (ed.) A. Erdélyi (ed.) et al. (ed.) , Higher transcendental functions , 2. Bessel functions, parabolic cylinder functions, orthogonal polynomials , McGraw-Hill (1953)
[JES]	E. Jahnke, F. Emde, F. Lösch, "Tafeln höheren Funktionen" , Teubner (1966)
[S]	H. Struve, Ann. Physik Chemie , 17 (1882) pp. 1008–1016
[Wa]	G.N. Watson, "A treatise on the theory of Bessel functions", 1–2, Cambridge Univ. Press (1952) MR1349110 MR1570252 MR0010746 MR1520278 Zbl 0849.33001 Zbl 0174.36202 Zbl 0063.08184