Si函数

Si 函数定义如下

File:Si(x) 2D plot.png

Si(x)的二维图像

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

<math>Si(z)</math>是下列三阶常微分方程的一个解:

<math>{\it Si} \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</math>

即：

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

表示为其他特殊函数

Meijer G-函数

• <math> </math>
• <math>\frac{ -1}{2}\,\sqrt {\pi }

G^{1, 1}_{1, 3}\left(1/4\,{z}^{2}\, \Big\vert\,^{1}_{1/2, 0, 0}\right) </math> 超几何函数

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

级数展开

• <math> {\it Si} \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

<math>Si(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:Si(x) Re complex 3D plot.png Si(x) Re complex 3D plot

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

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

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

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

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

参见

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