Shi函数

双曲正弦积分函数定义为

File:Shi(x) 2D plot.png

Shi(x) 2D plot

<math>{\it Shi} \left( z \right) =\int _{0}^{z}\!{\frac {\sinh \left( t \right) }{t}}{dt}</math>

<math>Shi(z)</math>是下列三阶常微分方程的一个解：

<math>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</math>

即：

<math>w \left( z \right) ={\it \_C1}+{\it \_C2}\,{\it Shi} \left( z \right) +{\it \_C3}\,{\it Chi} \left( z \right)</math>

与其他特殊函数的关系

Meijer G函数

• <math> </math>

超几何函数

• <math>Shi(z)=z*_{1}F_{2}(1/2;3/2, 3/2; (1/4)*z^2) </math>

• <math>\frac{ -1}{2}\,i\sqrt {\pi }

G^{1, 1}_{1, 3}\left(-1/4\,{z}^{2}\, \Big\vert\,^{1}_{1/2, 0, 0}\right)

   </math>

级数展开

• <math> {\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) )

   </math>

帕德近似

帕德近似 <math>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}</math>

图集

File:Shi(x) Re complex 3D plot.png Shi(x) Re complex 3D plot

File:Shi(x) Im complex 3D plot.png Shi(x) Im complex 3D plot

File:Shi(x) abs complex 3D plot.png Shi(x) abs complex 3D plot

File:Shi(x) abs complex density plot.JPG Shi(x) abs complex density plot

File:Shi(x) Re complex density plot.JPG Shi(x) Re complex density plot

File:Shi(x) Im complex density plot.JPG Shi(x) Im complex density plot

参见

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