Chi 函数 定义如下[ 1] [ 2]
Chi(x) 2D plot Chi(x) 3D plot
C
h
i
(
z
)
=
∫
0
z
cosh
(
t
)
t
d
t
{\displaystyle {\it {Chi}}\left(z\right)=\int _{0}^{z}\!{\frac {\cosh \left(t\right)}{t}}{dt}}
C
h
i
(
z
)
{\displaystyle Chi(z)}
z
d
d
z
w
(
z
)
−
2
d
2
d
z
2
w
(
z
)
−
z
d
3
d
z
3
w
(
z
)
=
0
{\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：
w
(
z
)
=
_
C
1
+
_
C
2
C
h
i
(
z
)
+
_
C
3
S
h
i
(
z
)
{\displaystyle w\left(z\right)={\it {\_C1}}+{\it {\_C2}}\,{\it {Chi}}\left(z\right)+{\it {\_C3}}\,{\it {Shi}}\left(z\right)}
C
h
i
(
−
z
)
=
C
h
i
(
z
)
{\displaystyle Chi(-z)=Chi(z)}
Meijer G函数
{\displaystyle }
−
1
2
π
G
1
,
3
2
,
0
(
−
1
/
4
z
2
|
0
,
0
,
1
/
2
1
)
−
1
/
2
i
π
{\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
C
h
i
(
z
)
=
z
∗
1
F
2
(
1
,
1
;
3
/
2
,
2
,
2
;
(
1
/
4
)
∗
z
2
)
{\displaystyle Chi(z)=z*_{1}F_{2}(1,1;3/2,2,2;(1/4)*z^{2})}
C
h
i
(
z
)
=
(
γ
+
ln
(
z
)
+
1
4
z
2
+
1
96
z
4
+
1
4320
z
6
+
1
322560
z
8
+
1
36288000
z
10
+
1
5748019200
z
12
+
1
1220496076800
z
14
+
O
(
z
16
)
)
{\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))}
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
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.
Sloane, N. J. A. Sequence A061079 in "The On-Line Encyclopedia of Integer Sequences
_2D_plot.png)
_3D_plot.png)
_Re_complex_3D_plot.png)
_Im_complex_3D_plot.png)
_abs_complex_3D_plot.png)
_abs_complex_density_plot.JPG)
_Re_complex_density_plot.JPG)
_Im_complex_density_plot.JPG)
