Shi函数

双曲正弦积分函数定义为^[1][2]

Shi(x) 2D plot

${\displaystyle {\it {Shi}}\left(z\right)=\int _{0}^{z}\!{\frac {\sinh \left(t\right)}{t}}{dt}}$

${\displaystyle Shi(z)}$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：

${\displaystyle w\left(z\right)={\it {\_C1}}+{\it {\_C2}}\,{\it {Shi}}\left(z\right)+{\it {\_C3}}\,{\it {Chi}}\left(z\right)}$

与其他特殊函数的关系

Meijer G函数

• ${\displaystyle }$

超几何函数

• ${\displaystyle Shi(z)=z*_{1}F_{2}(1/2;3/2,3/2;(1/4)*z^{2})}$ 

• ${\displaystyle {\frac {-1}{2}}\,i{\sqrt {\pi }}G_{1,3}^{1,1}\left(-1/4\,{z}^{2}\,{\Big \vert }\,_{1/2,0,0}^{1}\right)}$ 

级数展开

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

帕德近似

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

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

参考文献

1. Abramowitz, M. and Stegun, I. A. (Eds.). "Sine and Cosine Integrals." §5.2 inHandbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 231-233, 1972.
2. Sloane, N. J. A. Sequence A061079 in "The On-Line Encyclopedia of Integer Sequences