$$\begin{array}{ cccc }
\operatorname{Ei} : & \mathbb{R} & \longrightarrow & \mathbb{R} \\
& x & \longmapsto & \displaystyle \int_{-x}^\infty \hspace{-2mm} \frac{e^{-t}}{-t} \text{d}t
\end{array}$$
$$\begin{array}{ cccc }
\operatorname{E}_n : & \mathbb{R} & \longrightarrow & \mathbb{R} \\
& x & \longmapsto & \displaystyle \int_1^\infty \hspace{-2mm} \frac{e^{-xt}}{t^n} \text{d}t
\end{array}$$
$$\begin{array}{ cccc }
\operatorname{Ein} : & \mathbb{R} & \longrightarrow & \mathbb{R} \\
& x & \longmapsto & \displaystyle \int_0^x \hspace{-3mm} 1-e^{-t} \text{d}\ln(t)
\end{array}$$
$$ \operatorname{E}_1(x) = -\operatorname{Ei}(-x) $$
$$ \operatorname{E}_n(x) = \int_0^1 \hspace{-3mm} e^{-\frac{x}{s}}s^{n-2}\text{d}s $$
$$ \operatorname{E}_n(x) = \int_1^\infty \hspace{-2mm} \frac{e^{-x t}}{t^n}\text{d}t $$
$$ \operatorname{E}_1(x) = \int_x^\infty \hspace{-2mm} \frac{e^{-t}}{t} \text{d}t $$
$$ {\operatorname{E}_n}^{(1)}(x) = -\operatorname{E}_{n-1}(x) \iff \operatorname{E}_{n+1}(x) = \frac{e^{-x}-x\operatorname{E}_n(x)}{n} $$
Autor: Đɑvɪẟ Ƒernández-De la Cruʒ.
\operatorname{Ei} : & \mathbb{R} & \longrightarrow & \mathbb{R} \\
& x & \longmapsto & \displaystyle \int_{-x}^\infty \hspace{-2mm} \frac{e^{-t}}{-t} \text{d}t
\end{array}$$
$$\begin{array}{ cccc }
\operatorname{E}_n : & \mathbb{R} & \longrightarrow & \mathbb{R} \\
& x & \longmapsto & \displaystyle \int_1^\infty \hspace{-2mm} \frac{e^{-xt}}{t^n} \text{d}t
\end{array}$$
$$\begin{array}{ cccc }
\operatorname{Ein} : & \mathbb{R} & \longrightarrow & \mathbb{R} \\
& x & \longmapsto & \displaystyle \int_0^x \hspace{-3mm} 1-e^{-t} \text{d}\ln(t)
\end{array}$$
$$ \operatorname{E}_1(x) = -\operatorname{Ei}(-x) $$
$$ \operatorname{E}_n(x) = \int_0^1 \hspace{-3mm} e^{-\frac{x}{s}}s^{n-2}\text{d}s $$
$$ \operatorname{E}_n(x) = \int_1^\infty \hspace{-2mm} \frac{e^{-x t}}{t^n}\text{d}t $$
$$ \operatorname{E}_1(x) = \int_x^\infty \hspace{-2mm} \frac{e^{-t}}{t} \text{d}t $$
$$ {\operatorname{E}_n}^{(1)}(x) = -\operatorname{E}_{n-1}(x) \iff \operatorname{E}_{n+1}(x) = \frac{e^{-x}-x\operatorname{E}_n(x)}{n} $$
Autor: Đɑvɪẟ Ƒernández-De la Cruʒ.