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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Evaluation of the integral $ \smallint \sp{\infty }\sb{0}t\sp{2\sp{\alpha }-1}J\nu (\chi \surd (1+t\sp{2}))/(1+t\sp{2})\sp{\alpha +\beta -1}dt$


Author: Paul W. Schmidt
Journal: Math. Comp. 32 (1978), 265-269
MSC: Primary 33A35
MathSciNet review: 0457812
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Abstract | References | Similar Articles | Additional Information

Abstract: Methods are developed for evaluating the integral

$\displaystyle I_\nu ^{\alpha \beta }(x) = \int_0^\infty {\frac{{{t^{2\alpha - 1}}{J_\nu }(x\sqrt {1 + {t^2}} )}}{{{{(1 + {t^2})}^{\alpha + \beta - 1}}}}dt,} $

where $ {J_\nu }(t)$ is the Bessel function of the first kind and order $ \nu $, $ \alpha > 0$, $ \beta > 1/4$, and $ \nu $ is real. Only $ I_\nu ^{1/2,1}(x)$ and $ I_\nu ^{\alpha ,\nu /2 + 1 - \alpha }(x)$ are included in previously published tables of integrals of Bessel functions. The integrals $ I_1^{1/2,1/2}(x)$ and $ I_2^{1/2,1}(x)$ are used in a technique developed by I. S. Fedorova for calculating the diameter distribution of long circular cylinders from small-angle x-ray, light, or neutron scattering data.

The $ I_\nu ^{\alpha \beta }(x)$ are shown to be proportional to a G function. From this result, power series expansions and recurrence relations are developed for use in evaluating the $ I_\nu ^{\alpha \beta }(x)$. A convenient expression is obtained for the quantity required in Fedorova's method for computing diameter distributions.


References [Enhancements On Off] (What's this?)

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  • [2] I. S. FEDOROVA & J. COLLOID, Interface Sci., v. 59, 1977, pp. 100-101.
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  • [12] Ref. 11, p. 77, Eq. (12).

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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1978-0457812-4
PII: S 0025-5718(1978)0457812-4
Article copyright: © Copyright 1978 American Mathematical Society