Shi函数

双曲正弦积分函数定义为

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

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

${\displaystyle z\frac{d}{dz}w\left(z\right)-<!--\; -\; -->2\frac{d^2}{d{z}^2}w\left(z\right)-<!--\; -\; -->z\frac{d^3}{d{z}^3}w\left(z\right)=0}$

即：

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

Meijer G函数

超几何函数

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