Lommel function

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

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

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

$s_{\mu ,\nu }(z)=\frac{\pi }{2}(Y_{\nu }(z)\underset{0}{\overset{z}{\int }}{x}^{\mu }{J}_{\nu }(x)dx-J_{\nu }(z)\underset{0}{\overset{z}{\int }}{x}^{\mu }{Y}_{\nu }(x)dx),$

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

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

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The function $s_{\mu ,\nu }$ can also be written as

$s_{\mu ,\nu }(z)=\frac{z^{\mu +1}}{(\mu -\nu +1)(\mu +\nu +1)}{}_1{F}_2(1;\frac{\mu }{2}-\frac{\nu }{2}+\frac{3}{2},\frac{\mu }{2}+\frac{\nu }{2}+\frac{3}{2};-\frac{z^2}{4}),$

where ${}_1{F}_2$ is a generalized hypergeometric function.

See also

• Anger function
• Lommel polynomial
• Struve function
• Weber function

References

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

• "Ueber eine mit den Bessel'schen Functionen verwandte Function" (in de). Mathematische Annalen 9 (3): 425–444. 1875. doi:10.1007/BF01443342. https://zenodo.org/record/1568162. 

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

• Treatise on the Theory of Bessel Functions. Cambridge University Press. 1922. 

External links

• Paris, R. B.. "Lommel functions". https://dlmf.nist.gov/11.9. 

• Solomentsev, E. D.. "Lommel function". https://encyclopediaofmath.org/index.php?title=Lommel_function. 

• Weisstein, Eric W.. "Lommel differential equation". https://mathworld.wolfram.com/LommelDifferentialEquation.html. 

• Weisstein, Eric W.. "Lommel function". https://mathworld.wolfram.com/LommelFunction.html.