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A double integral containing the modified Bessel function: asymptotics and computation


Author: N. M. Temme
Journal: Math. Comp. 47 (1986), 683-691
MSC: Primary 33A40; Secondary 41A60, 65D30
DOI: https://doi.org/10.1090/S0025-5718-1986-0856712-X
MathSciNet review: 856712
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Abstract: A two-dimensional integral containing $ \exp ( - u - t){I_0}(2\sqrt {ut} )$ is considered. $ {I_0}(z)$ is the modified Bessel function and the integral is taken over the rectangle $ 0 \leqslant u \leqslant x$, $ 0 \leqslant t \leqslant y$. The integral is difficult to compute when x and y are large, especially when x and y are almost equal. Computer programs based on existing series expansions are inefficient in this case. A representation in terms of the error function (normal distribution function) is discussed, from which more efficient algorithms can be constructed.


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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1986-0856712-X
Keywords: Bessel function integral, uniform asymptotic expansion, incomplete gamma functions
Article copyright: © Copyright 1986 American Mathematical Society

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