# 指数积分

## 定义

${\displaystyle {\mbox{Ei}}(x)=\int _{-\infty }^{x}{\frac {e^{t}}{t}}\,\mathrm {d} t.\,}$

${\displaystyle {\rm {E}}_{1}(z)=\int _{z}^{\infty }{\frac {e^{-t}}{t}}\,\mathrm {d} t,\qquad |{\rm {Arg}}(z)|<\pi .}$

${\displaystyle {\rm {E}}_{1}(z)=\int _{1}^{\infty }{\frac {e^{-tz}}{t}}\,\mathrm {d} t,\qquad \Re (z)\geq 0.}$

Ei与E1有以下关系：

${\displaystyle {\rm {Ei}}(-x\pm {\rm {i}}0)=-{\rm {E}}_{1}(x)\mp {\rm {i}}\pi ,\quad ~~~~~~~~(x>0)}$
${\displaystyle -{\rm {Ei}}(x)={\frac {1}{2}}{\rm {E}}_{1}(-x+{\rm {i}}0)+{\frac {1}{2}}{\rm {E}}_{1}(-x-{\rm {i}}0),\qquad ~~~~~~~~(x>0)~.}$

## 性质

### 收敛级数

${\displaystyle {\mbox{Ei}}(x)=\gamma +\ln x+\sum _{k=1}^{\infty }{\frac {x^{k}}{k\;k!}}\,,~~~~~x>0}$
${\displaystyle E_{1}(z)=-\gamma -\ln z+\sum _{k=1}^{\infty }{\frac {(-1)^{k+1}z^{k}}{k\;k!}}\,,~~~~~~~~{\rm {Re}}(z)>0}$

### 渐近（发散）级数

${\displaystyle E_{1}(z)={\frac {\exp(-z)}{z}}\left[\sum _{n=0}^{N-1}{\frac {n!}{(-z)^{n}}}+{\mathcal {O}}\left({\frac {N!}{z^{N}}}\right)\right]}$

### 指数和对数的表现

${\displaystyle ~E_{1}~}$在自变量较大时的表现类似指数函数，自变量较小时类似对数函数。${\displaystyle ~E_{1}~}$是位于以下两个函数之间的：

${\displaystyle {\frac {\exp(-x)}{2}}\!~\ln \!\left(1+{\frac {2}{x}}\right)\!0}$

### 与其它函数的关系

li(x) = Ei (ln (x))    对于所有正实数x ≠ 1。

${\displaystyle {\rm {E}}_{1}(x)=\int _{1}^{\infty }{\frac {e^{-tx}}{t}}\,\mathrm {d} t=\int _{x}^{\infty }{\frac {e^{-t}}{t}}\,\mathrm {d} t.}$

${\displaystyle {\rm {Ei}}(-x)=-{\rm {E}}_{1}(x).\,}$

${\displaystyle {\rm {Ein}}(x)=\int _{0}^{x}(1-e^{-t})\,{\frac {\mathrm {d} t}{t}}=\sum _{k=1}^{\infty }{\frac {(-1)^{k+1}x^{k}}{k\;k!}}.}$

${\displaystyle {\rm {E}}_{1}(z)\,=\,-\gamma -\ln z+{\rm {Ein}}(z),~~~~~~|{\rm {Arg}}(z)|<\pi ~}$

${\displaystyle {\rm {Ei}}(x)\,=\,\gamma +\ln x-{\rm {Ein}}(-x),~~~~~~x>0.}$

${\displaystyle {\rm {E}}_{n}(x)=\int _{1}^{\infty }{\frac {e^{-xt}}{t^{n}}}\,\mathrm {d} t,}$

${\displaystyle {\rm {E}}_{n}(x)=x^{n-1}\Gamma (1-n,x).\,}$

${\displaystyle \varphi _{m}(x)={\rm {E}}_{-m}(x).\,}$

### 導數

${\displaystyle {{\rm {E}}_{n}}'(z){n-1}(z),~~~~~~~~(|{\rm {Arg}}(z)|<\pi ,~~~n>0)}$

### 複數變數指數積分

${\displaystyle {\rm {E}}_{1}(z)=\int _{1}^{\infty }{\frac {\exp(-zt)}{t}}\,{\rm {d}}t,~~~~~~({\rm {Re}}(z)\geq 0)}$

${\displaystyle {\rm {E}}_{1}({\rm {i}}\!~x)=-{\frac {\pi }{2}}+{\rm {Si}}(x)-{\rm {i}}\cdot {\rm {Ci}}(x),~~~~~~~~~(x>0)}$

## 参考文献

1. ^ Abramovitz, Milton; Irene Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Abramowitz and Stegun. New York: Dover. 1964 [2008-08-27]. ISBN 0-486-61272-4. （原始内容存档于2010-10-11）.
• R. D. Misra, Proc. Cambridge Phil. Soc. 36, 173 (1940)
• S. Chandrasekhar, Radiative transfer, reprinted 1960, Dover