Lommel function

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

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

Two solutions are given by the Lommel functions "s"_μ,ν("z") and "S"_μ,ν("z"), introduced by harvs|txt|authorlink=Eugen von Lommel|first=Eugen von|last= Lommel|year=1880,

: $s\_\{mu,\; u\}(z)\; =\; frac\{1\}\{2\}\; pi\; left\; [\; Y\_\; u\; (z)\; int\_0^z\; z^mu\; J\_\; u\; (z),\; dz\; -\; J\_\; u\; (z)\; int\_0^z\; z^mu\; Y\_\; u\; (z),\; dz\; ight]$: $displaystyle\; S\_\{mu,\; u\}(z)\; =\; s\_\{mu,\; u\}(z)\; -frac\{2^\{mu-1\}Gamma(frac\{1+mu+\; u\}\{2\})\}\{piGamma(frac\{\; u-mu\}\{2\})\}left(J\_\; u(z)-cos(pi(mu-\; u)/2)Y\_\; u(z)\; ight)$

where "J"_ν("z") is a Bessel function of the first kind, and "Y"_ν("z") a Bessel function of the second kind.

ee also

* Anger function
* Lommel polynomial
* Struve function
* Weber function

References

*Citation | last1=Erdélyi | first1=Arthur | last2=Magnus | first2=Wilhelm | author2-link=Wilhelm Magnus | last3=Oberhettinger | first3=Fritz | last4=Tricomi | first4=Francesco G. | title=Higher transcendental functions. Vol II | publisher=McGraw-Hill Book Company, Inc., New York-Toronto-London | id=MathSciNet | id = 0058756 | year=1953
*citation|first=E.|last= Lommel|title=Ueber eine mit den Bessel'schen Functionen verwandte Function|journal= Math. Ann. |volume= 9 |year=1875|pages= 425-444
*citation|first=E.|last= Lommel|title=Zur Theorie der Bessel'schen Funktionen IV|journal= Math. Ann. |volume= 16 |year=1880|pages= 183–208
*springer|id=l/l060800|first=E.D. |last=Solomentsev

External links

* Weisstein, Eric W. [http://mathworld.wolfram.com/LommelDifferentialEquation.html "Lommel Differential Equation."] From MathWorld--A Wolfram Web Resource.
* Weisstein, Eric W. [http://mathworld.wolfram.com/LommelFunction.html "Lommel Function."] From MathWorld--A Wolfram Web Resource.