Si 函数

Si 函数定义如下

${\displaystyle \mathit{S}\mathit{i}\left(z\right)={\int <!--\; \int \; -->}_0^z\frac{\sin <!--\; \; -->\left(t\right)}{t}dt}$

${\displaystyle Si(z)}$是下列三阶常微分方程的一个解:

${\displaystyle \mathit{S}\mathit{i}\left(z\right)=z\frac{d}{dz}w\left(z\right)+2\frac{d^2}{d{z}^2}w\left(z\right)+z\frac{d^3}{d{z}^3}w\left(z\right)=0}$

即：

${\displaystyle w\left(z\right)=\mathit{\_<!--\; \_\; -->}\mathit{C}\mathit{1}+\mathit{\_<!--\; \_\; -->}\mathit{C}\mathit{2}\mathit{S}\mathit{i}\left(z\right)+\mathit{\_<!--\; \_\; -->}\mathit{C}\mathit{3}\mathit{C}\mathit{i}\left(z\right)}$

Meijer G函数

超几何函数

${\displaystyle Si(z)\approx <!--\; \approx \; -->\left(-<!--\; -\; -->\frac{33317056220720070437}{9686419676455776844590000}{z}^7+\frac{67177799936189717}{98024149196718942600}{z}^5-<!--\; -\; -->\frac{540705278447237}{16111793096107650}{z}^3+z\right){\left(1+\frac{177197169001594}{8055896548053825}{z}^2+\frac{87368534024947}{363052404432292380}{z}^4+\frac{212787117226481}{131788022808922133940}{z}^6+\frac{10065927082366801}{1707972775603630855862400}{z}^8\right)}^{-<!--\; -\; -->1}}$