Struve function

Graph of $\mathrm {H} _{n}(x)$ for $n\in [0,1,2,3,4,5]$

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:

${\begin{aligned}\mathbf {H} _{\alpha -1}(x)+\mathbf {H} _{\alpha +1}(x)&={\frac {2\alpha }{x}}\mathbf {H} _{\alpha }(x)+{\frac {\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {3}{2}}\right)}},\\\mathbf {H} _{\alpha -1}(x)-\mathbf {H} _{\alpha +1}(x)&=2{\frac {d}{dx}}\left(\mathbf {H} _{\alpha }(x)\right)-{\frac {\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {3}{2}}\right)}}.\end{aligned}}$

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

${\begin{aligned}\mathbf {E} _{n}(z)&={\frac {1}{\pi }}\sum _{k=0}^{\left\lfloor {\frac {n-1}{2}}\right\rfloor }{\frac {\Gamma \left(k+{\frac {1}{2}}\right)\left({\frac {z}{2}}\right)^{n-2k-1}}{\Gamma \left(n-k-{\frac {1}{2}}\right)}}\mathbf {H} _{n}\\\mathbf {E} _{-n}(z)&={\frac {(-1)^{n+1}}{\pi }}\sum _{k=0}^{\left\lfloor {\frac {n-1}{2}}\right\rfloor }{\frac {\Gamma (n-k-{\frac {1}{2}})\left({\frac {z}{2}}\right)^{-n+2k+1}}{\Gamma \left(k+{\frac {3}{2}}\right)}}\mathbf {H} _{-n}.\end{aligned}}$

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 ₁F₂ (which is not the Gauss hypergeometric function ₂F₁):

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

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 Ann, eds. (1983) [June 1964]. "Chapter 12". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 496. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.
• Ivanov, A.B. (2001) [1994], "Struve function", Encyclopedia of Mathematics, EMS Press
• Paris, R. B. (2010), "Struve function", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
• Struve, H. (1882). "Beitrag zur Theorie der Diffraction an Fernröhren". Annalen der Physik und Chemie. 17 (13): 1008–1016. Bibcode:1882AnP...253.1008S. doi:10.1002/andp.18822531319.

External links

• Struve functions at the Wolfram functions site.