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The Sign of Lommel's Function

J. Steinig
Transactions of the American Mathematical Society
Vol. 163 (Jan., 1972), pp. 123-129
DOI: 10.2307/1995711
Stable URL: http://www.jstor.org/stable/1995711
Page Count: 7
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The Sign of Lommel's Function
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Abstract

Lommel's function $s_{\mu, v}(x)$ is a particular solution of the differential equation $x^2y'' + xy' + (x^2 - v^2)y = x^{\mu + 1}$. It is shown here that $s_{\mu, v}(x) > 0$ for $x > 0$, if $\mu = \frac{1}{2}$ and $|v| < \frac{1}{2}$, or if $\mu > \frac{1}{2}$ and $|v| \leqq \mu$. This includes earlier results of R. G. Cooke's. The sign of $s_{\mu, v}(x)$ for other values of $\mu$ and $v$ is also discussed.

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