Lommel function

The Lommel differential equation, named after Eugen von Lommel, is an inhomogeneous form of the Bessel differential equation:

z 2 d 2 y d z 2 + z d y d z + ( z 2 − ν 2 ) y = z μ + 1 . {\displaystyle z^{2}{\frac {d^{2}y}{dz^{2}}}+z{\frac {dy}{dz}}+(z^{2}-\nu ^{2})y=z^{\mu +1}.}

Solutions are given by the Lommel functions sμ,ν(z) and Sμ,ν(z), introduced by Eugen von Lommel (1880),

s μ , ν ( z ) = π 2 [ Y ν ( z ) ∫ 0 z x μ J ν ( x ) d x − J ν ( z ) ∫ 0 z x μ Y ν ( x ) d x ] , {\displaystyle s_{\mu ,\nu }(z)={\frac {\pi }{2}}\left[Y_{\nu }(z)\!\int _{0}^{z}\!\!x^{\mu }J_{\nu }(x)\,dx-J_{\nu }(z)\!\int _{0}^{z}\!\!x^{\mu }Y_{\nu }(x)\,dx\right],}
S μ , ν ( z ) = s μ , ν ( z ) + 2 μ − 1 Γ ( μ + ν + 1 2 ) Γ ( μ − ν + 1 2 ) ( sin ⁡ [ ( μ − ν ) π 2 ] J ν ( z ) − cos ⁡ [ ( μ − ν ) π 2 ] Y ν ( z ) ) , {\displaystyle S_{\mu ,\nu }(z)=s_{\mu ,\nu }(z)+2^{\mu -1}\Gamma \left({\frac {\mu +\nu +1}{2}}\right)\Gamma \left({\frac {\mu -\nu +1}{2}}\right)\left(\sin \left[(\mu -\nu ){\frac {\pi }{2}}\right]J_{\nu }(z)-\cos \left[(\mu -\nu ){\frac {\pi }{2}}\right]Y_{\nu }(z)\right),}

where Jν(z) is a Bessel function of the first kind and Yν(z) a Bessel function of the second kind.

The s function can also be written as[1]

s μ , ν ( z ) = z μ + 1 ( μ − ν + 1 ) ( μ + ν + 1 ) 1 F 2 ( 1 ; μ 2 − ν 2 + 3 2 , μ 2 + ν 2 + 3 2 ; − z 2 4 ) , {\displaystyle s_{\mu ,\nu }(z)={\frac {z^{\mu +1}}{(\mu -\nu +1)(\mu +\nu +1)}}{}_{1}F_{2}(1;{\frac {\mu }{2}}-{\frac {\nu }{2}}+{\frac {3}{2}},{\frac {\mu }{2}}+{\frac {\nu }{2}}+{\frac {3}{2}};-{\frac {z^{2}}{4}}),}

where pFq is a generalized hypergeometric function.

See also

  • Anger function
  • Lommel polynomial
  • Struve function
  • Weber function

References

  1. ^ Watson's "Treatise on the Theory of Bessel functions" (1966), Section 10.7, Equation (10)
  • Erdélyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz; Tricomi, Francesco G. (1953), Higher transcendental functions. Vol II (PDF), McGraw-Hill Book Company, Inc., New York-Toronto-London, MR 0058756
  • Lommel, E. (1875), "Ueber eine mit den Bessel'schen Functionen verwandte Function", Math. Ann., 9 (3): 425–444, doi:10.1007/BF01443342
  • Lommel, E. (1880), "Zur Theorie der Bessel'schen Funktionen IV", Math. Ann., 16 (2): 183–208, doi:10.1007/BF01446386
  • Paris, R. B. (2010), "Lommel function", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
  • Solomentsev, E.D. (2001) [1994], "Lommel function", Encyclopedia of Mathematics, EMS Press

External links

  • Weisstein, Eric W. "Lommel Differential Equation." From MathWorld—A Wolfram Web Resource.
  • Weisstein, Eric W. "Lommel Function." From MathWorld—A Wolfram Web Resource.