Calculus How To

Struve Function

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The Struve function is an uncommon special function defined by:
struve function definition

Where:

While the function definition might seem a little complicated, you’ll rarely have to solve it. Most popular mathematical software includes it as a built in function. For example, in Maple, the function is StruveH. There are a few online calculators, including this one that calculates the modified Struve function, Lv(x).

Notation for the Struve Function

The Struve function is generally denoted by Hv(x) or Hn(x). Occasionally you may see the “H” bolded (Hv(x)) or in lowercase hv(x), sometimes to distinguish it from Hermite polynomials. The modified Struve is usually denoted with an L instead of H.

Use as Solutions to Bessel Differential Equations

Developed by H. Struve in 1882, the function is a solution to the non-homogeneous (or inhomogeneous) Bessel differential equation. Two versions of the function exist; Each version solves a different form of the inhomogenous Bessel equation:

bessel and modified

The Struve function solves the inhomogenous Bessel equation (1). A modified Struve solves the Bessel equation on the bottom (2).


Struve functions are generalized by the hypergeometric and Meijer G functions.

Uses

The Struve function can be used for various calculations in mathematical physics. For example, to describe thin disk vibration and for various purposes in the theory of electromagnetism.

References

Luke, Y. (1969). Special Functions and Their Approximations:, Volume 2. Academic Press.
Maplesoft. StruveH, StruveL. Retrieved December 2, 2019 from: https://www.maplesoft.com/support/help/Maple/view.aspx?path=StruveH
Oldham, K. et al. (2010). An Atlas of Functions: with Equator, the Atlas Function Calculator. Springer Science and Business Media.
Rojas, S. et al. (2015). Learning SciPy for Numerical and Scientific Computing – Second Edition. Packt Publishing Ltd.

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Stephanie Glen. "Struve Function" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/struve-function/
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