# 对数积分

## 积分表示法

${\displaystyle \operatorname {li} (x)=\int _{0}^{x}{\frac {dt}{\ln(t)}}}$

${\displaystyle \operatorname {li} (x)=\lim _{\varepsilon \to 0}\left(\int _{0}^{1-\varepsilon }{\frac {dt}{\ln(t)}}+\int _{1+\varepsilon }^{x}{\frac {dt}{\ln(t)}}\right)}$

## 特殊值與欧拉对数积分

${\displaystyle \operatorname {Li} (x)=\operatorname {li} (x)-\operatorname {li} (2)}$

${\displaystyle \operatorname {Li} (x)=\int _{2}^{x}{\frac {dt}{\ln t}}}$

${\displaystyle \operatorname {li} (2)=-(\Gamma \left(0,-\ln 2\right)+i\,\pi )\sim 1.045163780117492784844588889194613136522615578151}$

## 级数表示法

${\displaystyle {\hbox{li}}(x)={\hbox{Ei}}(\ln(x))}$

${\displaystyle \operatorname {li} (e^{u})={\hbox{Ei}}(u)=\gamma +\ln u+\sum _{n=1}^{\infty }{u^{n} \over n\cdot n!}\quad {\text{for }}u\neq 0}$

${\displaystyle \operatorname {li} (x)=\gamma +\ln \ln x+{\sqrt {x}}\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}(\ln x)^{n}}{n!\;2^{n-1}}}\sum _{k=0}^{\lfloor (n-1)/2\rfloor }{\frac {1}{2k+1}}}$

## 渐近展开式

x → ∞，函数有以下的渐进表现：

${\displaystyle \operatorname {li} (x)={\mathcal {O}}\left({x \over \ln(x)}\right)}$

${\displaystyle \operatorname {li} (x)={\frac {x}{\ln x}}\sum _{k=0}^{\infty }{\frac {k!}{(\ln x)^{k}}}}$

${\displaystyle {\frac {\operatorname {li} (x)}{x/\ln x}}=1+{\frac {1}{\ln x}}+{\frac {2}{(\ln x)^{2}}}+{\frac {6}{(\ln x)^{3}}}+\cdots }$

## 数论中的重要性

${\displaystyle \pi (x)\sim \operatorname {Li} (x)}$