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Struve Function

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StruveH

The Struve function of complex order nu, denoted H_nu(z) or occasionally H_nu(z), is defined as

H_nu(z)=(1/2z)^(nu+1)sum_(k=0)^infty((-1)^k(1/2z)^(2k))/(Gamma(k+3/2)Gamma(k+nu+3/2)),
(1)

where Gamma(z) is the gamma function (Abramowitz and Stegun 1972, pp. 496-499), and powers of z use the principal branch for z!=0. For integer orders n>=-1, H_n(z) is an entire function. For Re[nu]>-1/2, Watson (1966, p. 338) gives the integral representation

H_nu(z)=(2(1/2z)^nu)/(Gamma(nu+1/2)Gamma(1/2))int_0^1(1-t^2)^(nu-1/2)sin(zt)dt.
(2)

The Struve function is implemented as StruveH[n, z].

The Struve function and its derivatives satisfy

H_(nu-1)(z)-H_(nu+1)(z)=2H_nu^'(z)-((1/2z)^nu)/(sqrt(pi)Gamma(nu+3/2)).
(3)

For integer n, H_n(z) is a particular solution of the differential equation

z^2y^('')+zy^'+(z^2-n^2)y=2/pi(z^(n+1))/((2n-1)!!),
(4)

where n!! 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)],
(5)

where

R_1(x)=1-(2J_1(x))/x
(6)
X_1(x)=(2H_1(x))/x
(7)

for x!=0, with R_1(0)=X_1(0)=0 defined by continuity. Here a is the piston radius, k is the wavenumber omega/c, rho is the density of the medium, c is the speed of sound, J_1(x) is the first order Bessel function of the first kind and H_1(z) is the Struve function of the first kind (Aarts and Janssen 2003, p. 2635).

StruveHReIm
StruveHContours

The illustrations above show the values of the Struve function H_0(z) in the complex plane.

For integer orders,

H_0(z)=2/pisum_(k=0)^(infty)((-1)^k)/([(2k+1)!!]^2)z^(2k+1)
(8)
=2/pi(z-1/9z^3+1/(225)z^5-1/(11025)z^7+1/(893025)z^9-...)
(9)
H_1(z)=2/pisum_(k=1)^(infty)((-1)^(k+1))/((2k-1)!!(2k+1)!!)z^(2k)
(10)
=2/pi(1/3z^2-1/(45)z^4+1/(1575)z^6-1/(99225)z^8+...)
(11)

(OEIS A001818 and A079484).

StruveH1Approximation

A simple approximation of H_1(x) for real number x is given by

H_1(x) approx h(x) =2/pi-J_0(x)+((16)/pi-5)(sinx)/x+(12-(36)/pi)(1-cosx)/(x^2),
(12)

with integrated squared approximation error on [0,infty) approximately 2.2×10^(-4) by Parseval's relation (Aarts and Janssen 2003, p. 2636). The right-hand side of equation (12) has the continuous extension h(0)=0=H_1(0). The maximum absolute error of the approximation is reported to be about 0.005. The maximum relative error appears to be less than 1% and decays to zero for x->infty (Aarts and Janssen 2003, pp. 2636-2637).

Aarts and Janssen (2016, pp. 4154-4155) developed improved approximations to H_0(z) and H_1(z) for z>=0. These are obtained by least squares fitting of the auxiliary function sqrt((1-t)/(1+t)) on [0,1] with a continuous piecewise linear function having two pieces. Recurrence relations extend the approximations to higher integer orders (Aarts and Janssen 2016, p. 4159).

For half-integer orders with n a nonnegative integer and z!=0,

H_(n+1/2)(z)=Y_(n+1/2)(z)+1/pisum_(k=0)^(n)(Gamma(k+1/2)(1/2z)^(-2k+n-1/2))/(Gamma(n+1-k))
(13)
H_(-(n+1/2))(z)=(-1)^nJ_(n+1/2)(z).
(14)

The first few cases are

H_(1/2)(z)=sqrt(2/(piz))(1-cosz)
(15)
H_(3/2)(z)=(2+z^2-2cosz-2zsinz)/(sqrt(2pi)z^(3/2))
(16)
H_(5/2)(z)=(24+4z^2+z^4+8(z^2-3)cosz-24zsinz)/(4sqrt(2pi)z^(5/2)).
(17)

See also

Anger Function, Bessel Function, Modified Struve Function, Weber Functions

Related Wolfram sites

https://functions.wolfram.com/Bessel-TypeFunctions/StruveH/

Portions of this entry contributed by Ronald M. Aarts

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References

Aarts, R. M. and Janssen, A. J. E. M. "Approximation of the Struve Function H_1 Occurring in Impedance Calculations." J. Acoust. Soc. Amer. 113, 2635-2637, 2003. https://doi.org/10.1121/1.1564019.Aarts, R. M. and Janssen, A. J. E. M. "Efficient Approximation of the Struve Functions H_n Occurring in the Calculation of Sound Radiation Quantities." J. Acoust. Soc. Amer. 140, 4154-4160, 2016. https://doi.org/10.1121/1.4968792.Abramowitz, M. "Tables of Integrals of Struve Functions." J. Math. Phys. 29, 49-51, 1950.Abramowitz, M. and Stegun, I. A. (Eds.). "Struve Function H_nu(x)." §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 H_nu(x) and L_nu(x)." J. Math. Anal. Appl. 137, 17-36, 1989. https://doi.org/10.1016/0022-247X(89)90270-9.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 L_nu(z) and H_nu(z)." J. Math. Phys. 25, 252-259, 1946.Prudnikov, A. P.; Marichev, O. I.; and Brychkov, Yu. A. "The Struve Functions H_nu(x) and L_nu(x)." §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.

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Struve Function

Cite this as:

Weisstein, Eric W., with contributions by Ronald M. Aarts. "Struve Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/StruveFunction.html

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