Struve function

In mathematics, the Struve functions, are solutions of the non-homogeneous Bessel's differential equation:
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 as, where is the Neumann function.
The modified Struve functions are equal to and are solutions of the non-homogeneous Bessel's differential equation:
And further defined its second-kind version as, where is the modified Bessel function.

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
where is the gamma function.
The modified Struve functions, denoted, have the following power series form

Integral form

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

Asymptotic forms

For small, the power series expansion is given [|above].
For large, one obtains:
where is the Neumann function.

Properties

The Struve functions satisfy the following recurrence relations:

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
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
where the right hand side is a spherical Bessel function.
Struve functions can be expressed in terms of the generalized hypergeometric function :

Applications

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