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))],](https://mathworld.wolfram.com/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],](https://mathworld.wolfram.com/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