Chi函数

Chi 函数定义如下^[1][2]

Chi(x) 2D plot

Chi(x) 3D plot

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

${\displaystyle Chi(z)}$ 是下列三阶非线性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：

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

对称性

${\displaystyle Chi(-z)=Chi(z)}$ 

表示为其他特殊函数

Meijer G函数

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

超几何函数

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

级数展

• ${\displaystyle {\it {Chi}}\left(z\right)=(\gamma +\ln \left(z\right)+{\frac {1}{4}}{z}^{2}+{\frac {1}{96}}{z}^{4}+{\frac {1}{4320}}{z}^{6}+{\frac {1}{322560}}{z}^{8}+{\frac {1}{36288000}}{z}^{10}+{\frac {1}{5748019200}}{z}^{12}+{\frac {1}{1220496076800}}{z}^{14}+O\left({z}^{16}\right))}$ 

图集

Chi(x) Re complex 3D plot

Chi(x) Im complex 3D plot

Chi(x) abs complex 3D plot

Chi(x) abs complex density plot

Chi(x) Re complex density plot

Chi(x) Im complex density plot

参见

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