Struve Function

In mathematics, the Struve functions , are solutions of the non-homogeneous Bessel's differential equation:

: $x^2 \frac + x \frac + \left (x^2 - \alpha^2 \right )y = \frac$

introduced by . The complex number α is the order of the Struve function, and is often an integer.

And further defined its second-kind version $\mathbf_\alpha(x)$ as $\mathbf_\alpha(x)=\mathbf_\alpha(x)-Y_\alpha(x)$.

The modified Struve functions are equal to , are solutions of the non-homogeneous Bessel's differential equation:

: $x^2 \frac + x \frac - \left (x^2 + \alpha^2 \right )y = \frac$

And further defined its second-kind version $\mathbf_\alpha(x)$ as $\mathbf_\alpha(x)=\mathbf_\alpha(x)-I_\alpha(x)$.

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 have the power series form

:$\mathbf_\alpha(z) = \sum_^\infty \frac \left(\right)^,$

where is the gamma function.

The modified Struve functions, denoted , have the following power series form

:$\mathbf_\alpha(z) = \sum_^\infty \frac \left(\frac\right)^.$

Integral form

Another definition of the Struve function, for values of satisfying , is possible expressing in term of the Poisson's integral representation:

$\mathbf_\alpha(x)=\frac\int_0^1(1-t^2)^\sin xt~dt=\frac\int_0^\frac\sin(x\cos\tau)\sin^\tau~d\tau=\frac\int_0^\frac\sin(x\sin\tau)\cos^\tau~d\tau$

$\mathbf_\alpha(x)=\frac\int_0^\infty(1+t^2)^e^~dt=\frac\int_0^\infty e^\cosh^\tau~d\tau$

$\mathbf_\alpha(x)=\frac\int_0^1(1-t^2)^\sinh xt~dt=\frac\int_0^\frac\sinh(x\cos\tau)\sin^\tau~d\tau=\frac\int_0^\frac\sinh(x\sin\tau)\cos^\tau~d\tau$

$\mathbf_\alpha(x)=-\frac\int_0^1(1-t^2)^e^~dt=-\frac\int_0^\frace^\sin^\tau~d\tau=-\frac\int_0^\frace^\cos^\tau~d\tau$

Asymptotic forms

For small , the power series expansion is given above.

For large , one obtains:

:$\mathbf_\alpha(x) - Y_\alpha(x) = \frac + O\left(\left (\tfrac\right)^\right),$

where is the Neumann function.

Properties

The Struve functions satisfy the following recurrence relations:

:$\begin \mathbf_(x) + \mathbf_(x) &= \frac \mathbf_\alpha (x) + \frac, \\ \mathbf_(x) - \mathbf_(x) &= 2 \frac \left (\mathbf_\alpha(x) \right) - \frac. \end$

Relation to other functions

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

:$\begin \mathbf_n(z) &= \frac \sum_^ \frac -\mathbf_n(z),\\ \mathbf_(z) &= \frac\sum_^ \frac-\mathbf_(z). \end$

Struve functions of order where is an integer can be expressed in terms of elementary functions. In particular if is a non-negative integer then

:$\mathbf_ (z) = (-1)^n J_(z),$

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 :

:$\mathbf_(z) = \frac _1F_2 \left (1;\tfrac, \alpha+\tfrac;-\tfrac \right ).$

Applications

The Struve and Weber functions were shown to have an application to beamforming in.K. Buchanan, C. Flores, S. Wheeland, J. Jensen, D. Grayson and G. Huff, "Transmit beamforming for radar applications using circularly tapered random arrays," 2017 IEEE Radar Conference (RadarConf), 2017, pp. 0112-0117, doi: 10.1109/RADAR.2017.7944181.

References

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External links

Struve functions
a
the Wolfram functions site

{{DEFAULTSORT:Struve Function
Special functions
Struve family