The Struve function, denoted 
or occasionally 
, is defined as
 |
(1) |
where 
is the gamma function (Abramowitz and Stegun 1972,
pp. 496-499). Watson (1966, p. 338) defines the Struve function as
 |
(2) |
The Struve function is implemented as StruveH[n, z].
The Struve function and its derivatives satisfy
 |
(3) |
For integer 
,
the Struve function gives the solution to
 |
(4) |
where 
is the double factorial.
The Struve function arises in the problem of the rigid-piston radiator mounted in an infinite baffle, which has radiation impedance given by
![Z=rhocpia^2[R_1(2ka)-iX_1(2ka)],](https://txtdot.deep-swarm.xyz/proxy/img?url=https%3A%2F%2Fmathworld.wolfram.com%2Fimages%2Fequations%2FStruveFunction%2FNumberedEquation5.svg){kind=small} |
(5) |
where
where 
is the piston radius, 
is the wavenumber 
,
is the density of the medium, 
is the speed of sound, 
is the first order Bessel
function of the first kind and 
is the Struve function of the first kind.
The illustrations above show the values of the Struve function 
in the complex plane.
For integer orders,
A simple approximation of 
for real 
is given by
 |
(12) |
with squared approximation error on 
equal to 
by Parseval's formula (Aarts and Janssen 2003).
The right-hand side of equation (12) equals 
for 
. The approximation error is small and decently spread-out
over the whole 
-range,
vanishes for 
,
and reaches its maximum value at about 0.005. The maximum relative error appears
to be less than 1% and decays to zero for 
.
For half integer orders,
The first few cases are
See also
Anger Function, Bessel Function, Modified Struve Function, Weber Functions
Related Wolfram sites
http://functions.wolfram.com/Bessel-TypeFunctions/StruveH/
Portions of this entry contributed by Ronald M. Aarts
Explore with Wolfram|Alpha
References
Aarts, R. M. and Janssen, A. J. E. M. "Approximation of the Struve Function 
Occurring in Impedance Calculations." J. Acoust.
Soc. Amer. 113, 2635-2637, 2003.Abramowitz, M. "Tables
of Integrals of Struve Functions." J. Math. Phys. 29, 49-51, 1950.Abramowitz,
M. and Stegun, I. A. (Eds.). "Struve Function 
." §12.1 in Handbook
of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, pp. 496-498, 1972.Apelblat, A. "Derivatives
and Integrals with Respect to the Order of the Struve Functions 
and 
." J. Math. Anal. Appl. 137, 17-36,
1999.Cook, R. K. "Some Properties of Struve Functions."
J. Washington Acad. Sci. 47, 365-368, 1957.Horton, C. W.
"On the Extension of Some Lommel Integrals to Struve Functions with an Application
to Acoustic Radiation." J. Math. Phys. 29, 31-37, 1950.Horton,
C. W. "A Short Table of Struve Functions and of Some Integrals Involving
Bessel and Struve Functions." J. Math. Phys. 29, 56-58, 1950.Mathematical
Tables Project. "Table of the Struve Functions 
and 
." J. Math. Phys. 25, 252-259, 1946.Prudnikov,
A. P.; Marichev, O. I.; and Brychkov, Yu. A. "The Struve Functions
and 
." §1.4 in Integrals
and Series, Vol. 3: More Special Functions. Newark, NJ: Gordon and Breach,
pp. 24-27, 1990.Sloane, N. J. A. Sequences A001818/M4669
and A079484 in "The On-Line Encyclopedia
of Integer Sequences."Spanier, J. and Oldham, K. B. "The
Struve Function." Ch. 57 in An
Atlas of Functions. Washington, DC: Hemisphere, pp. 563-571, 1987.Watson,
G. N. A
Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge
University Press, 1966.
Referenced on Wolfram|Alpha
Cite this as:
Aarts, Ronald M. and Weisstein, Eric W. "Struve Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/StruveFunction.html