Lommel Function
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
![画像:(_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))],](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/LommelFunction/Inline9.svg){kind=link}
where _1F_2 and _0F_1 are generalized and confluence hypergeometric functions, respectively and s_(mu,nu)^((1))(z) is typically denoted just as s_(mu,nu)(z).
The function S_(mu,nu)(z) defined by Gradshteyn and Ryzhik (2000, p. 936) is identical to s_(mu,nu)^((2))(z).
The Lommel functions s_(mu,nu)^((1))(z) and s_(mu,nu)^((2))(z) are implemented in the Wolfram Language as LommelS1 [m, n, z] and LommelS2 [m, n, z], respectively.
s_(mu,nu)(z) is also given by
![画像: 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],](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/LommelFunction/NumberedEquation1.svg){kind=link}
where J_nu(z) and Y_nu(z) are Bessel functions of the first and second kinds (Watson 1966, p. 346; Gradshteyn and Ryzhik 2000, pp. 936-937).
If a minus sign precedes the z^(mu+1) term in the general form of Lommel differential equation, then the solution is
where K_nu(z) and I_nu(z) are modified Bessel functions of the first and second kinds. These functions are closely related to the modified Lommel functions.
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),
These functions were first defined by Lommel (1884-1886ab). Note that the definition (7) of V_n(w,z) differs by a factor of (-1)^n from the modern convention (Watson 1966, p. 537) and from the definition of Born and Wolf (1989, p. 438).
See also
Lommel Differential Equation, Lommel Polynomial, Modified Lommel FunctionExplore with Wolfram|Alpha
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References
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 Bruxelles 31, 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 geradlinig begrenzter Schirme." Abh. der math. phys. Classe der k. b. Akad. der Wiss. (München) 15, 529-664, 1884-1886b.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.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." Nature 71, 607-608, 1905.Prudnikov, A. P.; Marichev, O. I.; and Brychkov, Yu. A. "The Lommel Functions s_(mu,nu)(x) and S_(mu,nu)(x)." §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, England: 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.Referenced on Wolfram|Alpha
Lommel FunctionCite this as:
Weisstein, Eric W. "Lommel Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LommelFunction.html