Lommel Differential Equation

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

: $z^2 \frac + z \frac + (z^2 - \nu^2)y = z^.$

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 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

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*{{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