imported>Procyon117: v2.05 - Fix errors for CW project (Reference list missing)

2024-05-10T16:05:00Z

v2.05 - Fix errors for CW project (Reference list missing)

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[[File:LommelS1.png|thumb]]
[[File:LommelS2.png|thumb]]
The '''Lommel differential equation''', named after [[Eugen von Lommel]], is an inhomogeneous form of the [[Bessel differential equation]]:

: <math>z^2 \frac{d^2y}{dz^2} + z \frac{dy}{dz} + (z^2 - \nu^2)y = z^{\mu+1}.</math>

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}},

:<math>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],</math>
:<math>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),</math>

where ''J''ν(''z'') is a [[Bessel function]] of the first kind and ''Y''ν(''z'') a Bessel function of the second kind.

The ''s'' function can also be written as<ref>Watson's "Treatise on the Theory of Bessel functions" (1966), Section 10.7, Equation (10)</ref>
:<math> 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}),</math>
where ''p''''F''''q'' is a [[generalized hypergeometric function]].

==See also==
* [[Anger function]]
* [[Lommel polynomial]]
* [[Struve function]]
* [[Weber function (disambiguation)|Weber function]]

==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]]

Lommel function - Revision history

imported>Procyon117: v2.05 - Fix errors for CW project (Reference list missing)