# 对数积分

## 积分表示法

${\rm li} (x) = \int_{0}^{x} \frac{dt}{\ln (t)} \; .$

${\rm li} (x) = \lim_{\varepsilon \to 0} \left( \int_{0}^{1-\varepsilon} \frac{dt}{\ln (t)} + \int_{1+\varepsilon}^{x} \frac{dt}{\ln (t)} \right) \; .$

## 特殊值與欧拉对数积分

${\rm Li}(x) = {\rm li}(x) - {\rm li}(2) \,$

${\rm Li} (x) = \int_{2}^{x} \frac{dt}{\ln t} \,$

${\rm li} (2) = -(\Gamma\left(0,-\ln 2\right) + i\,\pi) \sim 1.04516 37801 17492 78484 45888 89194 613136 522615 578151$

## 级数表示法

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

${\rm li} (e^{u}) = \hbox{Ei}(u) = \gamma + \ln u + \sum_{n=1}^{\infty} {u^{n}\over n \cdot n!} \quad {\rm for} \; u \ne 0 \; ,$

${\rm li} (x) = \gamma + \ln \ln x + \sqrt{x} \sum_{n=1}^{\infty} \frac{ (-1)^{n-1} (\ln x)^n} {n! \, 2^{n-1}} \sum_{k=0}^{\lfloor (n-1)/2 \rfloor} \frac{1}{2k+1} .$

## 渐近展开式

x → ∞，函数有以下的渐进表现：

${\rm li} (x) = \mathcal{O} \left( {x\over \ln (x)} \right) \; .$

$\rm li} (x) = \frac{x}{\ln x} \sum_{k=0}^{\infnty\frac{k!}{(\ln x)^k$

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

## 数论中的重要性

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