Lommel function

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In mathematics, the Lommel differential equation, named after Eugen von Lommel, is an inhomogeneous form of the Bessel differential equation

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

Its solutions are given by the Lommel functions ${\displaystyle s_{\mu ,\nu }(z)}$ and ${\displaystyle S_{\mu ,\nu }(z)}$ :^[1]

${\displaystyle 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\$

${\displaystyle S_{\mu ,\nu }(z)=s_{\mu ,\nu }(z)+2^{\mu -1}\Gamma \left({\frac {\mu +\nu +1}{2\right)\Gamma \left({\frac {\mu -\nu +1}{2\right)\left(\sin \left[(\mu -\nu ){\frac {\pi }{2\right]$

where ${\displaystyle J_{\nu }(z)}$ is a Bessel function of the first kind and ${\displaystyle Y_{\nu }(z)}$ a Bessel function of the second kind.

The function ${\displaystyle s_{\mu ,\nu$ can also be written as^[2]

${\displaystyle s_{\mu ,\nu }(z)={\frac {z^{\mu +1{(\mu -\nu +1)(\mu +\nu +1){}_{1}F_{2}{\bigg (}1;{\frac {\mu }{2-{\frac {\nu }{2+{\frac {3}{2,{\frac {$

where ${\displaystyle {}_{1}F_{2$ is a generalized hypergeometric function.

References#edit

^ Lommel (1880).
^ Watson (1922), p. 346, equation 10.

Erdélyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz; Tricomi, Francesco G. (1953). Higher Transcendental Functions. Vol. I, II. New York: McGraw-Hill. MR 0058756.

Lommel, E. (1875). "Ueber eine mit den Bessel'schen Functionen verwandte Function". Mathematische Annalen (in German). 9 (3): 425–444. doi:10.1007/BF01443342.

Lommel, E. (1880). "Zur Theorie der Bessel'schen Functionen". Mathematische Annalen (in German). 16 (2): 183–208. doi:10.1007/BF01446386.

Watson, G. N. (1922). Treatise on the Theory of Bessel Functions. Cambridge University Press. JFM 48.0412.02.

External links#edit

Paris, R. B. "Lommel functions". Digital Library of Mathematical Functions.

Solomentsev, E. D. "Lommel function". Encyclopedia of Mathematics.

Weisstein, Eric W. "Lommel differential equation". MathWorld.

Weisstein, Eric W. "Lommel function". MathWorld.

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Lommel function

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