https://en.formulasearchengine.com/w/api.php?action=feedcontributions&feedformat=atom&user=130.161.0.0%2F16formulasearchengine - User contributions [en]2026-09-17T18:34:21ZUser contributionsMediaWiki 1.47.0-wmf.7https://en.formulasearchengine.com/w/index.php?title=Sod_shock_tube&diff=18088Sod shock tube2013-03-22T08:58:33Z<p>130.161.17.254: </p>
<hr />
<div>The '''Lommel differential equation''' is an inhomogeneous form of the [[Bessel differential equation]]:<br />
<br />
:<math>z^2 \frac{d^2y}{dz^2} + z \frac{dy}{dz} + (z^2 - \nu^2)y = z^{\mu+1}.</math><br />
<br />
Two solutions are given by the '''Lommel functions''' ''s''<sub>μ,ν</sub>(''z'') and ''S''<sub>μ,ν</sub>(''z''), introduced by {{harvs|txt|authorlink=Eugen von Lommel|first=Eugen von|last= Lommel|year=1880}},<br />
<br />
: <math>s_{\mu,\nu}(z) = \frac{1}{2} \pi \left[ Y_\nu (z) \int_0^z z^\mu J_\nu (z)\, dz - J_\nu (z) \int_0^z z^\mu Y_\nu (z)\, dz\right]</math><br />
: <math>\displaystyle S_{\mu,\nu}(z) = s_{\mu,\nu}(z) -\frac{2^{\mu-1}\Gamma(\frac{1+\mu+\nu}{2})}{\pi\Gamma(\frac{\nu-\mu}{2})}<br />
\left(J_\nu(z)-\cos(\pi(\mu-\nu)/2)Y_\nu(z)\right)</math><br />
<br />
where ''J''<sub>ν</sub>(''z'') is a [[Bessel function]] of the first kind, and ''Y''<sub>ν</sub>(''z'') a Bessel function of the second kind.<br />
<br />
==See also==<br />
* [[Anger function]]<br />
* [[Lommel polynomial]]<br />
* [[Struve function]]<br />
* [[Weber function]]<br />
<br />
==References==<br />
*{{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}}<br />
*{{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}}<br />
*{{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}}<br />
*{{dlmf|id=11.9|first=R. B. |last=Paris}}<br />
*{{springer|id=l/l060800|first=E.D. |last=Solomentsev}}<br />
<br />
==External links==<br />
* Weisstein, Eric W. [http://mathworld.wolfram.com/LommelDifferentialEquation.html "Lommel Differential Equation."] From MathWorld—A Wolfram Web Resource.<br />
* Weisstein, Eric W. [http://mathworld.wolfram.com/LommelFunction.html "Lommel Function."] From MathWorld—A Wolfram Web Resource.<br />
<br />
[[Category:Special functions]]</div>130.161.17.254https://en.formulasearchengine.com/w/index.php?title=Tinkerbell_map&diff=243072Tinkerbell map2011-10-17T12:35:53Z<p>130.161.210.162: /* Source Code */</p>
<hr />
<div>Hi there. Let me start by introducing the writer, her title is Sophia. For years she's been living in Kentucky but her husband wants them to move. Playing badminton is a thing that he is totally addicted to. Credit authorising is how he makes money.<br><br>Feel free to surf to my web-site ... [http://www.seekavideo.com/playlist/2199/video/ real psychic readings]</div>130.161.210.162