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## 8.2 The Exponential Integral Function and Functions Generated by It

### 8.21 The exponential integral function Ei(x)

8.211

1.

$\begin{array}{ll}\text{Ei(}x\text{)=}-{\int }_{-x}^{\infty }\frac{{e}^{-t}}{t}\text{d}t={\int }_{-\infty }^{x}\frac{{e}^{t}}{t}\text{d}t=\text{li}\left({e}^{x}\right)\hfill & \hfill \left[\mathrm{Re}x>0\right]\end{array}$

2.11

$\begin{array}{ll}\text{Ei(}x\text{)=}-\underset{\epsilon \to 0+}{\mathrm{lim}}\left[{\int }_{-x}^{-\epsilon }\frac{{e}^{-t}}{t}\text{d}t={\int }_{\epsilon }^{\infty }\frac{{e}^{-t}}{t}\text{d}t\right]=\text{PV}{\int }_{-\infty }^{x}\frac{{e}^{t}}{t}\hfill & \text{d}t\hfill \\ \left[x>0\right]\hfill \end{array}$

3.7

$\begin{array}{ll}\text{Ei(}x\text{)=}\frac{1}{2}\left\{\text{Ei(}x+i0\text{)+Ei(}x-i0\text{)}\right\}\hfill & \hfill \left[x>0\right]\end{array}$

ET I 386

8.212

1.8

$\begin{array}{ll}\text{Ei(}-x\text{)=}C+\mathrm{ln}x+{\int }_{0}^{x}\frac{{e}^{-t}-1}{t}\text{d}t\hfill & \left[x>0\right]\hfill \end{array}$

NT 11(1)

$\begin{array}{l}\text{=}C+{e}^{-x}\mathrm{ln}x+{\int }_{0}^{x}\hfill \end{array}$

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