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)}
是下列三阶非线性常微分方程的一个解:
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)}
对称性
C
h
i
(
−
z
)
=
C
h
i
(
z
)
{\displaystyle Chi(-z)=Chi(z)}
表示为其他特殊函数
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})}
级数展
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))}
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
参考文献
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