Shi函数

双曲正弦积分函数定义为

Shi(x) 2D plot

$𝑆ℎ𝑖\left(z\right)=\underset{0}{\overset{z}{\int }}\frac{\sinh \left(t\right)}{t}dt$

$Shi(z)$是下列三阶常微分方程的一个解：

$z\frac{d}{dz}w\left(z\right)-2\frac{d^2}{d{z}^2}w\left(z\right)-z\frac{d^3}{d{z}^3}w\left(z\right)=0$

即：

$w\left(z\right)=\mathrm{\_}𝐶1+\mathrm{\_}𝐶2𝑆ℎ𝑖\left(z\right)+\mathrm{\_}𝐶3𝐶ℎ𝑖\left(z\right)$

与其他特殊函数的关系

Meijer G函数

• ${\displaystyle }$

超几何函数

• $Shi(z)=z{*}_1{F}_2(1/2;3/2,3/2;(1/4)*z^2)$

• $\frac{-1}{2}i\sqrt{\pi }{G}_{1,3}^{1,1}(-1/4z^2|{}_{1/2,0,0}^1)$

级数展开

• $𝑆ℎ𝑖\left(z\right)=(z+\frac{1}{18}{z}^3+\frac{1}{600}{z}^5+\frac{1}{35280}{z}^7+\frac{1}{3265920}{z}^9+\frac{1}{439084800}{z}^{11}+\frac{1}{80951270400}{z}^{13}+O\left(z^{15}\right))$

帕德近似

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

图集

Shi(x) Re complex 3D plot

Shi(x) Im complex 3D plot

Shi(x) abs complex 3D plot

Shi(x) abs complex density plot

Shi(x) Re complex density plot

Shi(x) Im complex density plot

参见

• Sinhc函数
• Coshc函数
• Tanc函数
• Tanhc函数
• Chi函数