Struve Functions

Mini Article

New additions to the NAG Library include a set of six routines related to Struve functions. These are:

s17ga – Struve function of order $0$ , $H_{0} (x)$
s17gb – Struve function of order $1$ , $H_{1} (x)$
s18ga – Modified Struve function of order $0$ , $L_{0} (x)$
s18gb – Modified Struve function of order $1$ , $L_{1} (x)$
s18gc – Modified Bessel function - modified Struve function, both order zero, $I_{0} (x) - L_{0} (x)$
s18gd – Modified Bessel function - modified Struve function, both first order, $I_{1} (x) - L_{1} (x)$

Applications

Struve functions have some specific uses across many different fields of physics in a wide variety of applications. For example, they can be found in water-wave and surface-wave problems (specifically flow of liquid near a turning ship) as well as calculations to do with the distribution of fluid pressure over a vibrating disk and other unsteady aerodynamics. They also crop up when considering aspects of optical diffraction, plasma stability (specifically resistive magnetohydrodynamics instability theory), quantum dynamical studies of spin decoherence and excitation in carbon nanotubes.

Defining Struve functions and their related functions

Struve functions are solutions of the non-homogeneous Bessel’s differential equation:

x^{2} \frac{d^{2} y}{d x^{2}} + x \frac{d y}{d x} + (x^{2} - α^{2}) y = \frac{4 {(\frac{x}{2})}^{α + 1}}{\sqrt{π} Γ (α + \frac{1}{2})}

and are defined as:

$H_{α} (x) = \frac{2 {(\frac{x}{2})}^{α}}{Γ (α + \frac{1}{2}) Γ (\frac{1}{2})} \int_{0}^{\frac{π}{2}} \sin (x \cos θ) \sin^{2 α} (θ) d θ$

Modified Struve functions are defined as:

$L_{α} (x) = I_{- α} (x) - \frac{2 {(\frac{x}{2})}^{α}}{Γ (α + \frac{1}{2}) Γ (\frac{1}{2})} \int_{0}^{\infty} \sin (x u) {(1 + u^{2})}^{α - \frac{1}{2}} d u$

where $I (x)$ is the modified Bessel function, defined as:

$I_{α} (x) = {(\frac{1}{2} x)}^{α} \sum_{k = 0}^{\infty} \frac{{(\frac{1}{4} x^{2})}^{k}}{k! Γ (α + k + 1)}$

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