Table of Integrals, Series, and Products, 8th Edition

8.2 The Exponential Integral Function and Functions Generated by It

8.21 The exponential integral function Ei(x)

8.211

1. 

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

2.¹¹ 

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

3.⁷ 

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

  ET I 386

8.212

1.⁸ 

$\begin{array}{ll}\text{Ei(}-x\text{)=}C+\ln x+{\displaystyle {\int }_0^x\frac{e^{-t}-1}{t}\text{d}t}\hfill & [x>0]\hfill \end{array}$

  NT 11(1)

$\begin{array}{l}\text{=}C+e^{-x}\ln x+{\displaystyle {\int }_0^x}\hfill \end{array}$