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Lommel function

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

: z^2 \frac{d^2y}{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 {{harvs|txt|authorlink=Eugen von Lommel|first=Eugen von|last= Lommel|year=1880}},

: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_{\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 asWatson's "Treatise on the Theory of Bessel functions" (1966), Section 10.7, Equation (10)

: s_{\mu, \nu} (z) = \frac{z^{\mu + 1}}{(\mu - \nu + 1)(\mu + \nu + 1)} {}_1F_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

References

{{reflist}}

  • {{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 | mr=0058756 | year=1953 | url=http://apps.nrbook.com/bateman/Vol2.pdf}}
  • {{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|doi=10.1007/BF01443342|issue=3|url= https://zenodo.org/record/1568162}}
  • {{citation|first=E.|last= Lommel|title=Zur Theorie der Bessel'schen Funktionen IV|journal= Math. Ann. |volume= 16 |year=1880|pages= 183–208|doi=10.1007/BF01446386|issue=2}}
  • {{dlmf|id=11.9|first=R. B. |last=Paris}}
  • {{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.

Category:Special functions

Category:Ordinary differential equations