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)],](/images/equations/StruveFunction/NumberedEquation5.svg) |
(5)
|
where
 |  |  |
(6)
|
 |  |  |
(7)
|
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,
 |  | ![2/pisum_(k=0)^(infty)((-1)^k)/([(2k+1)!!]^2)z^(2k+1)](/images/equations/StruveFunction/Inline22.svg) |
(8)
|
 |  |  |
(9)
|
 |  |  |
(10)
|
 |  |  |
(11)
|
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,
 |  |  |
(13)
|
 |  |  |
(14)
|
The first few cases are
 |  |  |
(15)
|
 |  |  |
(16)
|
 |  |  |
(17)
|
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 
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