Chi函数

Chi 函数定义如下

File:Chi(x) 2D plot.png

Chi(x) 2D plot

File:Chi(x) 3D plot.png

Chi(x) 3D plot

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

<math>Chi(z)</math> 是下列三阶非线性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 Chi} \left( z \right) +{\it \_C3}\,{\it Shi} \left( z \right)</math>

对称性

<math>Chi(-z)=Chi(z)</math>

表示为其他特殊函数

Meijer G函数

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

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

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

级数展开

• <math> {\it Chi} \left( z \right) =(\gamma+\ln \left( z \right) +{\frac {1}{4}}{z}^{2}+{\frac {1}{96}}{z}^{4}+{\frac {1}{4320}}{z}^{6}+{\frac {1}{322560}}{z}^{8}+{\frac {1}{

36288000}}{z}^{10}+{\frac {1}{5748019200}}{z}^{12}+{\frac {1}{ 1220496076800}}{z}^{14}+O \left( {z}^{16} \right) ) </math>

图集

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

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

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

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

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

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

参见

• Sinhc函数
• Coshc函数
• Tanc函数
• Tanhc函数
• Shi函数