The Exponential Integral Ei(x) and Numerical Calculation of Its Inverse

Abstract

The exponential integral y = Ei(x) = dx represents a transcendental function that can be applied as a growth function with interesting properties. Representing an infinite series, its numerical evaluation can be carried out with assymptotic expansion or summing up a finite number of elements with a sufficiently small remainder. Its inverse results in two subfunctions that are defined here as x = (y) for x < 0 (and always y < 0), and x = (y) for x > 0, where x is a real-valued variable. There is no direct formula to calculate numerical values for the two inverse subfunctions. In this article, we first show that for x being close to 0, the inverse function with sufficiently small error is represented by Lambert's W-function. Then we compare two methods. In the first method, two versions of a generic root-finding algorithm search for a number X that solves Y = Ei(X) for given number Y, distinguishing x < 0 and x > 0 in the case of Y < 0. The second method is a Taylor expansion with a 10th-degree polynomial. For the Y-value of interest, a corresponding reference interval is taken by the algorithm from a list that is here developed, and the associated X₀ together with Y enters the polynomial to calculate X directly. The chosen remainder or maximum error ε_Y , that results from cutting the infinite Taylor series, is taken into account in the interval limits around X₀. The relationship between ε_Y and corresponding ε_X (the resulting maximum error for X) is analyzed. Iterative formulas for the determination of optimal intervals as a function of X₀ are derived for given |ε_Y| or relative error . The methods were tested with Mathematica notebooks that are provided as supporting information. The more flexible root-finding algorithm will generally be the best choice. However, the Taylor expansion can be implemented without any special functions, as long as the corresponding polynomial can be solved accurately.