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Transactions of the American Mathematical Society

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The sign of Lommel's function


Author: J. Steinig
Journal: Trans. Amer. Math. Soc. 163 (1972), 123-129
MSC: Primary 33.25
DOI: https://doi.org/10.1090/S0002-9947-1972-0284625-X
MathSciNet review: 0284625
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Abstract: Lommel's function $ {s_{\mu ,\nu }}(x)$ is a particular solution of the differential equation $ {x^2}y'' + xy' + ({x^2} - {\nu ^2})y = {x^{\mu + 1}}$. It is shown here that $ {s_{\mu ,\nu }}(x) > 0$ for $ x > 0$, if $ \mu = \tfrac{1}{2}$ and $ \vert\nu \vert < \tfrac{1}{2}$, or if $ \mu > \tfrac{1}{2}$ and $ \vert\nu \vert \leqq \mu $. This includes earlier results of R. G. Cooke's. The sign of $ {s_{\mu ,\nu }}(x)$ for other values of $ \mu $ and $ \nu $ is also discussed.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0284625-X
Keywords: Lommel functions, Bessel functions, changes of sign, oscillation theorems, inhomogeneous Bessel equation
Article copyright: © Copyright 1972 American Mathematical Society

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