Contents
Exponential Integral
 Introduction
 Equation EI.1.1
 Series and asymptotic expansions
 Asymptotic expansion at $0$
 Asymptotic expansion at $\infty$
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EI Exponential Integral

EI.1 Introduction

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

\begin{equation*} \begin{split} (1 - x) \frac{\partial y (x)}{\partial x} + x \frac{\partial^{2} y (x)}{\partial x^{2}}& =0. \end{split} \end{equation*}
EI.1.1

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

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EI.2.1 Asymptotic expansion at $0$

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EI.2.1.1 First terms

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\begin{equation*} \begin{split} & \operatorname{Ei} (x)\approx \Biggl(-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} - \operatorname{ln} (x) + \gamma\ldots\Biggr). \end{split} \end{equation*} EI.2.1.1.1

EI.2.1.2 General form

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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 $\infty$

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EI.2.2.1 First terms

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\begin{equation*} \begin{split} & \operatorname{Ei} (x)\approx \operatorname{e} ^{\frac{1}{x}} x y _{0} (x), \end{split} \end{equation*}
where
\begin{equation*} \begin{split} y _{0} (x)& =1 + x + 2 x^{2} + 6 x^{3} + 2 \ldots \end{split} \end{equation*}

EI.2.2.2 General form

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EI.2.2.2.1 Auxiliary function $y _{0} (x)$

The coefficients $u (n)$ of $y _{0} (x)$ satisfy the following recurrence
\begin{equation*} \begin{split} -u (n) n + u (n - 1) \bigl(-1 + 2 n + (n - 1)^{2}\bigr)& =0 \end{split} \end{equation*}
whose initial conditions are given by
\begin{equation*} \begin{split} u (0)& =1 \end{split} \end{equation*}
This recurrence has the closed form solution
\begin{equation*} \begin{split} u (n)& =\Gamma (n + 1). \end{split} \end{equation*}
 
 
 
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