There are several functions called "Lommel functions." One type of Lommel function appear in the solution to the Lommel
differential equation and are given by
(1)
(2)
(3)
where
and
are generalized and confluence hypergeometric functions, respectively and is typically denoted just as .
The function
defined by Gradshteyn and Ryzhik (2000, p. 936) is identical to .
The Lommel functions
and
will be implemented in a future version of the Wolfram
Language as LommelS1[m, n, z] and LommelS2[m,
n, z], respectively.
Lommel functions of two variables are related to the Bessel function of the first kind and arise in the theory of diffraction (Chandrasekhar
1960, p. 369) and, in particular, Mie scattering (Watson 1966, p. 537),
(6)
(7)
These functions were first defined by Lommel (1884-1886ab). Note that the definition (7) of differs by a factor of from the modern convention (Watson 1966, p. 537)
and from the definition of Born and Wolf (1989, p. 438).
Born, M. and Wolf, E. Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of
Light, 6th ed. New York: Pergamon Press, 1989.Chandrasekhar,
S. Radiative
Transfer. New York: Dover, p. 369, 1960.Gilbert, L. P.
"Recherches analytiques sur la diffraction de la lumière." Mém.
courmonnées de l'Acad. R. des Sci. de Bruxelles31, 1-52, 1863.Gradshteyn,
I. S. and Ryzhik, I. M. Tables
of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press,
pp. 936-937, 2000.Gray, A. and Mathews, G. B. Ch. 14
in A
Treatise on Bessel Functions and Their Applications to Physics, 2nd ed. New
York: Dover, 1966.Hardy, G. H. "General Theorems in Contour
Integration with Some Applications." Quart. J.32, 369-384, 1901.Hardy,
G. H. "On Certain Definite Integrals Whose Values Can be Expressed in Terms
of Bessel's Functions." Messenger Math.38, 129-132, 1909.Lommel,
E. C. J. von. "Die Beugungserscheinungen einer kreisrunden Oeffnung
und eines kreisrunden Schirmchens theoretisch und experimentell bearbeitet."
Abh. der math. phys. Classe der k. b. Akad. der Wiss. (München)15,
229-328, 1884-1886a.Lommel, E. C. J. von. "Die Beugungserscheinungen
geradlinig begrenzter Schirme." Abh. der math. phys. Classe der k. b. Akad.
der Wiss. (München)15, 529-664, 1884-1886b.Mayall,
R. H. D. "On the Diffraction Pattern near the Focus of a Telescope."
Proc. Cambridge Philos. Soc.9, 259-269, 1898.Pocklington,
H. C. "Growth of a Wave-Group When the Group-Velocity Is Negative."
Nature71, 607-608, 1905.Prudnikov, A. P.; Marichev,
O. I.; and Brychkov, Yu. A. "The Lommel Functions and ." §1.5 in Integrals
and Series, Vol. 3: More Special Functions. Newark, NJ: Gordon and Breach,
pp. 28-29, 1990.Schafheitlin, F. "Beziehungen zwischen dem
Integrallogarithmus und den Besselschen Funktionen." Berliner Sitzungsber.8,
62-67, 1909.Szymanski, P. "On the Integral Representations of the
Lommel Functions." Proc. London Math. Soc.40, 71-82, 1936.Walker,
J. The
Analytical Theory of Light. London: Cambridge University Press, p. 396,
1904.Watson, G. N. §16.5-16.59 in A
Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge
University Press, pp. 537-550, 1966.
![(_1F_2(1;1/2(mu-nu+3),1/2(mu+nu+3);1/4z^2))/((mu+1)^2-nu^2)z^(mu+1)+(2^(mu-1)pi^2csc(pinu))/(Gamma(1/2(-mu-nu+1))Gamma(1/2(-mu+nu+1))[J_(-nu)(z)sec(1/2pi(mu+nu))-J_nu(z)sec(1/2pi(mu-nu))],](/images/equations/LommelFunction/Inline9.svg)
and 
are generalized and confluence hypergeometric functions, respectively and 
is typically denoted just as 
.
defined by Gradshteyn and Ryzhik (2000, p. 936) is identical to 
.
and 
will be implemented in a future version of the Wolfram
Language as LommelS1[m, n, z] and LommelS2[m,
n, z], respectively.
is also given by
![s_(mu,nu)(z)=1/2pi[Y_nu(z)int_0^zz^muJ_nu(z)dz-J_nu(z)int_0^zz^muY_nu(z)dz],](/images/equations/LommelFunction/NumberedEquation1.svg)
and 
are Bessel functions of the first
and second kinds (Watson 1966,
p. 346; Gradshteyn and Ryzhik 2000, pp. 936-937).
term in the general form of Lommel
differential equation, then the solution is
and 
are modified Bessel functions
of the first and second
kinds. These functions are closely related to the modified
Lommel functions.
differs by a factor of 
from the modern convention (Watson 1966, p. 537)
and from the definition of Born and Wolf (1989, p. 438).
and 
." §1.5 in