Computation of Exponential Integrals

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Abstract
Formulas leading to the computatmn of M member sequences of exponential integrals EN+k(x), x _> 0, N -> 1, k = 0, 1 .... M - 1, are presented here and nnplemented in Fortran subroutine EXPINT. Sequences of exponential integrals can be generated in a numerically stable fashion if recurrence is carrmd forward or backward away from the integer closest to x. In keeping with t his requirement, we select n, the integer closest to x within the constraint N - n -~ N + M - 1, and use E,(x) to start the recursion E,(x) Is computed by means of the power series on 0 ~ x ": 2 and t he confluent hypergeometnc function U(n, n, x) on 2 < x < o0. U(n, n, x) is, in turn, computed from the backward recurswe Miller algorithm for U(n + k, n, x), k = 0, 1, ..., with a normalizing relatmn derived from the two-term recursmn relation satisfied by E,(x) and E,+l(x). Truncation error bounds are derived and used m error tests m EXPINT Exponential scaling is also provided as a subroutine optmn.

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