Struve function

In

:

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

introduced by

α is the order of the Struve function, and is often an integer.

And further defined its second-kind version ${\displaystyle \mathbf {K} _{\alpha }(x)}$ as ${\displaystyle \mathbf {K} _{\alpha }(x)=\mathbf {H} _{\alpha }(x)-Y_{\alpha }(x)}$.

The modified Struve functions L_α(x) are equal to −ie^{−iαπ / 2}H_α(ix), are solutions y(x) of the non-homogeneous

Bessel's differential equation

:

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

And further defined its second-kind version ${\displaystyle \mathbf {M} _{\alpha }(x)}$ as ${\displaystyle \mathbf {M} _{\alpha }(x)=\mathbf {L} _{\alpha }(x)-I_{\alpha }(x)}$.

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(z) have the power series form

${\displaystyle \mathbf {H} _{\alpha }(z)=\sum _{m=0}^{\infty }{\frac {(-1)^{m}}{\Gamma \left(m+{\frac {3}{2}}\right)\Gamma \left(m+\alpha +{\frac {3}{2}}\right)}}\left({\frac {z}{2}}\right)^{2m+\alpha +1},}$

where Γ(z) is the gamma function.

The modified Struve functions, denoted L_α(z), have the following power series form

${\displaystyle \mathbf {L} _{\alpha }(z)=\sum _{m=0}^{\infty }{\frac {1}{\Gamma \left(m+{\frac {3}{2}}\right)\Gamma \left(m+\alpha +{\frac {3}{2}}\right)}}\left({\frac {z}{2}}\right)^{2m+\alpha +1}.}$

Integral form

Another definition of the Struve function, for values of α satisfying Re(α) > − 1/2, is possible expressing in term of the Poisson's integral representation:

Asymptotic forms

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

For large x, one obtains:

${\displaystyle \mathbf {H} _{\alpha }(x)-Y_{\alpha }(x)={\frac {\left({\frac {x}{2}}\right)^{\alpha -1}}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}+O\left(\left({\tfrac {x}{2}}\right)^{\alpha -3}\right),}$

where Y_α(x) is the Neumann function.

Properties

The Struve functions satisfy the following recurrence relations:

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

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

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

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

${\displaystyle \mathbf {H} _{-n-{\frac {1}{2}}}(z)=(-1)^{n}J_{n+{\frac {1}{2}}}(z),}$

where the right hand side is a

.

Struve functions (of any order) can be expressed in terms of the generalized hypergeometric function ₁F₂:

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

Applications

The Struve and Weber functions were shown to have an application to beamforming in.^[1], and in describing the effect of confining interface on Brownian motion of colloidal particles at low Reynolds numbers.^[2]

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.
  PMID 12765381
  .
• R. M. Aarts and Augustus J. E. M. Janssen (2016). "Efficient approximation of the Struve functions H_n occurring in the calculation of sound radiation quantities". J. Acoust. Soc. Am. 140 (6): 4154–4160.
  PMID 28040027
  .
• LCCN 65-12253
  .
• Ivanov, A. B. (2001) [1994], "Struve function", Encyclopedia of Mathematics, EMS Press
• Paris, R. B. (2010), "Struve function", in
  MR 2723248
  .
• Struve, H. (1882). "Beitrag zur Theorie der Diffraction an Fernröhren". Annalen der Physik und Chemie. 17 (13): 1008–1016.
  doi:10.1002/andp.18822531319
  .

External links

• Struve functions at the Wolfram functions site.