Lommel function

The Lommel differential equation is an inhomogeneous form of the Bessel differential equation:

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

Two solutions are given by the Lommel functions s_μ,ν(z) and S_μ,ν(z), introduced by Eugen von Lommel (1880),

$s_{\mu ,\nu }(z)={\frac {1}{2}}\pi \left[Y_{\nu }(z)\int _{0}^{z}z^{\mu }J_{\nu }(z)\,dz-J_{\nu }(z)\int _{0}^{z}z^{\mu }Y_{\nu }(z)\,dz\right]$

$\displaystyle S_{\mu ,\nu }(z)=s_{\mu ,\nu }(z)-{\frac {2^{\mu -1}\Gamma ({\frac {1+\mu +\nu }{2}})}{\pi \Gamma ({\frac {\nu -\mu }{2}})}}\left(J_{\nu }(z)-\cos(\pi (\mu -\nu )/2)Y_{\nu }(z)\right)$

where J_ν(z) is a Bessel function of the first kind, and Y_ν(z) a Bessel function of the second kind.

See also

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

References

• Erdélyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz; Tricomi, Francesco G. (1953), Higher transcendental functions. Vol II, McGraw-Hill Book Company, Inc., New York-Toronto-London, MR0058756
• Lommel, E. (1875), "Ueber eine mit den Bessel'schen Functionen verwandte Function", Math. Ann., 9: 425–444
• Lommel, E. (1880), "Zur Theorie der Bessel'schen Funktionen IV", Math. Ann., 16: 183–208
• 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.