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Modified Lommel Function


The modified Lommel functions of the first and second kind give the solution to the Lommel differential equation with a minus sign in front of the linear term, i.e.,

 z^2y^('')+zy^'-(z^2+n^2)y=z^(m+1).

They are denoted t_(m,n)(z) and T_(m,n)(z), respectively.

These functions will implemented in a future version of the Wolfram Language as LommelT1[m, n, z] and LommelT2[m, n, z], respectively.

Not that while Rollinger (1964) uses the notation R, T is preferable so as not to conflict with the notation for Lommel polynomials.


See also

Lommel Differential Equation, Lommel Function

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References

Rollinger, C. N. "Lommel Functions with Imagniary Argument." Quart. J. Appl. Math. 21, 343-349, 1964.Shafer, R. E. "Lommel Functions of Imaginary Arguments." Technical Report UCRL-7806. Livermore, CA: Lawrence Livermore National Lab, 1964.Szymanski, P. "On the Integral Representations of the Lommel Functions." Proc. London Math. Soc. 40, 71-82, 1936.Ziener, C. H. and Schlemmer, H. P. "the Inverse Laplace Transform of the Modified Lommel Functions." Integral Transforms and Special Functions 24, 141-155, 2013.

Cite this as:

Weisstein, Eric W. "Modified Lommel Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ModifiedLommelFunction.html

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