# 三角积分

## 正弦积分

${\displaystyle {\rm {Si}}(x)=\int _{0}^{x}{\frac {\sin t}{t}}\,dt}$
${\displaystyle {\rm {si}}(x)=-\int _{x}^{\infty }{\frac {\sin t}{t}}\,dt}$

${\displaystyle {\rm {Si}}(x)\,}$${\displaystyle {\frac {\sin x}{x}}\,}$的原函数，当${\displaystyle x=0\,}$时为零；${\displaystyle {\rm {si}}(x)\,}$${\displaystyle {\frac {\sin x}{x}}\,}$的原函数，当${\displaystyle x=\infty }$时为零。我们有：

${\displaystyle {\rm {si}}(x)={\rm {Si}}(x)-{\frac {\pi }{2}}}$

## 余弦积分

${\displaystyle {\rm {Ci}}(x)=\gamma +\ln x+\int _{0}^{x}{\frac {\cos t-1}{t}}\,dt}$
${\displaystyle {\rm {ci}}(x)=-\int _{x}^{\infty }{\frac {\cos t}{t}}\,dt}$
${\displaystyle {\rm {Cin}}(x)=\int _{0}^{x}{\frac {1-\cos t}{t}}\,dt}$

${\displaystyle {\rm {ci}}(x)\,}$${\displaystyle {\frac {\cos x}{x}}}$的原函数，当${\displaystyle x\to \infty }$时为零。我们有：

${\displaystyle {\rm {ci}}(x)={\rm {Ci}}(x)\,}$
${\displaystyle {\rm {Cin}}(x)=\gamma +\ln x-{\rm {Ci}}(x)\,}$

## 双曲正弦积分

${\displaystyle {\rm {Shi}}(x)=\int _{0}^{x}{\frac {\sinh t}{t}}\,dt={\rm {shi}}(x).}$

## 双曲余弦积分

${\displaystyle {\rm {Chi}}(x)=\gamma +\ln x+\int _{0}^{x}{\frac {\cosh t-1}{t}}\,dt={\rm {chi}}(x)}$

## 展开式

### 渐近展开式

${\displaystyle {\rm {Si}}(x)={\frac {\pi }{2}}-{\frac {\cos x}{x}}\left(1-{\frac {2!}{x^{2}}}+...\right)-{\frac {\sin x}{x}}\left({\frac {1}{x}}-{\frac {3!}{x^{3}}}+...\right)}$
${\displaystyle {\rm {Ci}}(x)={\frac {\sin x}{x}}\left(1-{\frac {2!}{x^{2}}}+...\right)-{\frac {\cos x}{x}}\left({\frac {1}{x}}-{\frac {3!}{x^{3}}}+...\right)}$

### 收敛级数

${\displaystyle {\rm {Si}}(x)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{(2n+1)(2n+1)!}}=x-{\frac {x^{3}}{3!\cdot 3}}+{\frac {x^{5}}{5!\cdot 5}}-{\frac {x^{7}}{7!\cdot 7}}\pm \cdots }$
${\displaystyle {\rm {Ci}}(x)=\gamma +\ln x+\sum _{n=1}^{\infty }{\frac {(-1)^{n}x^{2n}}{2n(2n)!}}=\gamma +\ln x-{\frac {x^{2}}{2!\cdot 2}}+{\frac {x^{4}}{4!\cdot 4}}\mp \cdots }$

## 与指数积分的关系

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

${\displaystyle {\rm {E}}_{1}({\rm {i}}\!~x)=i\left(-{\frac {\pi }{2}}+{\rm {Si}}(x)\right)-{\rm {Ci}}(x)=i~{\rm {si}}(x)-{\rm {ci}}(x)\qquad (x>0)}$