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 are implemented in 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 FunctionExplore with Wolfram|Alpha
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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.Referenced on Wolfram|Alpha
Modified Lommel FunctionCite this as:
Weisstein, Eric W. "Modified Lommel Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ModifiedLommelFunction.html