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The Lommel differential equation, named after
Eugen von Lommel Eugen Cornelius Joseph von Lommel (19 March 1837, Edenkoben – 19 June 1899, Munich) was a German physicist. He is notable for the Lommel polynomial, the Lommel function, the Lommel–Weber function, and the Lommel differential equation. He ...
, is an inhomogeneous form of the
Bessel differential equation Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary ...
: : z^2 \frac + z \frac + (z^2 - \nu^2)y = z^. Solutions are given by the Lommel functions ''s''μ,ν(''z'') and ''S''μ,ν(''z''), introduced by , :s_(z) = \frac \left Y_ (z) \! \int_^ \!\! x^ J_(x) \, dx - J_\nu (z) \! \int_^ \!\! x^ Y_(x) \, dx \right :S_(z) = s_(z) + 2^ \Gamma\left(\frac\right) \Gamma\left(\frac\right) \left(\sin \left \mu - \nu)\frac\rightJ_\nu(z) - \cos \left \mu - \nu)\frac\rightY_\nu(z)\right), where ''J''ν(''z'') is a
Bessel function Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary ...
of the first kind and ''Y''ν(''z'') a Bessel function of the second kind.


See also

*
Anger function In mathematics, the Anger function, introduced by , is a function defined as : \mathbf_\nu(z)=\frac \int_0^\pi \cos (\nu\theta-z\sin\theta) \,d\theta and is closely related to Bessel functions. The Weber function (also known as Lommel–Weber f ...
*
Lommel polynomial A Lommel polynomial ''R'm'',ν(''z''), introduced by , is a polynomial in 1/''z'' giving the recurrence relation :\displaystyle J_(z) = J_\nu(z)R_(z) - J_(z)R_(z) where ''J''ν(''z'') is a Bessel function of the first kind. They are given ...
*
Struve function In mathematics, the Struve functions , are solutions of the non-homogeneous Bessel's differential equation: : x^2 \frac + x \frac + \left (x^2 - \alpha^2 \right )y = \frac introduced by . The complex number α is the order of the Struve functi ...
* Weber function


References

* * * * *{{springer, id=l/l060800, first=E.D. , last=Solomentsev


External links

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

From MathWorld—A Wolfram Web Resource. Special functions