EI Exponential Integral

EI.1 Introduction

Let x be a complex variable of ℂ \ {0, ∞}.The function Exponential Integral (noted Ei) is defined by the following second order differential equation

\begin{aligned}& (1 − x)\frac{∂y(x)}{∂x} + x\frac{∂^{2}y(x)}{∂x^{2}} & = 0. \\ (EI.1.1) \end{aligned}

The initial conditions of EI.1.1 at 0 are not simple to state, since 0 is a (regular) singular point.

EI.2 Series and asymptotic expansions

EI.2.1 Asymptotic expansion at 0. EI.2.1.1 First terms.

\begin{matrix}Ei(x) ≈ (−x − \frac{x^{2}}{4} − \frac{x^{3}}{18} − \frac{x^{4}}{96} − \frac{x^{5}}{600} − \frac{x^{6}}{4320} − \frac{x^{7}}{35280} − \frac{x^{8}}{322560} − ln(x) + γ …). \\ (EI.2.1.1.1) \end{matrix}

EI.2.1.2 General form. The general form of is not easy to state and requires to exhibit the basis of formal solutions of ?? (coming soon).

EI.2.2 Asymptotic expansion at ∞. EI.2.2.1 First terms.

Ei(x) ≈ e^{\frac{1}{x}} xy_{0}(x),

where

y_{0}(x) = 1 + x + 2x^{2} + 6x^{3} + 2 …

EI.2.2.2 General form. EI.2.2.2.1 Auxiliary function y_{0}(x). The coefficients u(n) of y_{0}(x) satisfy the

following recurrence

−u(n)n + u(n − 1)(−1 + 2n + (n − 1)^{2}) = 0

whose initial conditions are given by

u(0) = 1

This recurrence has the closed form solution

u(n) = Γ(n + 1).