Struve function

In mathematics, the Struve functions Hα(x), are solutions y(x) of the non-homogeneous Bessel's differential equation: introduced by Hermann Struve (1882).

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

And further defined its second-kind version

The modified Struve functions Lα(x) are equal to −ie−iαπ / 2Hα(ix) and are solutions y(x) of the non-homogeneous Bessel's differential equation: And further defined its second-kind version

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.

Struve functions, denoted as Hα(z) have the power series form where Γ(z) is the gamma function.

The modified Struve functions, denoted Lα(z), have the following power series 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:

sin ⁡ ( x cos ⁡ τ )

⁡ τ

d τ =

sin ⁡ ( x sin ⁡ τ )

⁡ τ

d τ

− x sinh ⁡ τ

⁡ τ

d τ

sinh ⁡ ( x cos ⁡ τ )

⁡ τ

d τ =

sinh ⁡ ( x sin ⁡ τ )

⁡ τ

d τ

− x cos ⁡ τ

− x sin ⁡ τ

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

For large x, one obtains: where Yα(x) is the Neumann function.

The Struve functions satisfy the following recurrence relations: Struve functions of integer order can be expressed in terms of Weber functions En and vice versa: if n is a non-negative integer then 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 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: 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.