Chi 函數 定義如下[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)}
對稱性 [ 編輯 ]
C
h
i
(
−
z
)
=
C
h
i
(
z
)
{\displaystyle Chi(-z)=Chi(z)}
表示為其他特殊函數 [ 編輯 ] Meijer G函數
{\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 }
超幾何函數
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})}
級數展 [ 編輯 ]
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
參考文獻 [ 編輯 ]
^ 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)