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

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.