雙曲正弦積分函數 定義為[ 1] [ 2]
Shi(x) 2D plot
S
h
i
(
z
)
=
∫
0
z
sinh
(
t
)
t
d
t
{\displaystyle {\it {Shi}}\left(z\right)=\int _{0}^{z}\!{\frac {\sinh \left(t\right)}{t}}{dt}}
S
h
i
(
z
)
{\displaystyle Shi(z)}
常微分方程 的一個解：
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
S
h
i
(
z
)
+
_
C
3
C
h
i
(
z
)
{\displaystyle w\left(z\right)={\it {\_C1}}+{\it {\_C2}}\,{\it {Shi}}\left(z\right)+{\it {\_C3}}\,{\it {Chi}}\left(z\right)}
Meijer G函數
{\displaystyle }
超幾何函數
S
h
i
(
z
)
=
z
∗
1
F
2
(
1
/
2
;
3
/
2
,
3
/
2
;
(
1
/
4
)
∗
z
2
)
{\displaystyle Shi(z)=z*_{1}F_{2}(1/2;3/2,3/2;(1/4)*z^{2})}
−
1
2
i
π
G
1
,
3
1
,
1
(
−
1
/
4
z
2
|
1
/
2
,
0
,
0
1
)
{\displaystyle {\frac {-1}{2}}\,i{\sqrt {\pi }}G_{1,3}^{1,1}\left(-1/4\,{z}^{2}\,{\Big \vert }\,_{1/2,0,0}^{1}\right)}
S
h
i
(
z
)
=
(
z
+
1
18
z
3
+
1
600
z
5
+
1
35280
z
7
+
1
3265920
z
9
+
1
439084800
z
11
+
1
80951270400
z
13
+
O
(
z
15
)
)
{\displaystyle {\it {Shi}}\left(z\right)=(z+{\frac {1}{18}}{z}^{3}+{\frac {1}{600}}{z}^{5}+{\frac {1}{35280}}{z}^{7}+{\frac {1}{3265920}}{z}^{9}+{\frac {1}{439084800}}{z}^{11}+{\frac {1}{80951270400}}{z}^{13}+O\left({z}^{15}\right))}
帕德近似
S
h
i
(
z
)
≈
(
33317056220720070437
9686419676455776844590000
z
7
+
67177799936189717
98024149196718942600
z
5
+
540705278447237
16111793096107650
z
3
+
z
)
(
1
−
177197169001594
8055896548053825
z
2
+
87368534024947
363052404432292380
z
4
−
212787117226481
131788022808922133940
z
6
+
10065927082366801
1707972775603630855862400
z
8
)
−
1
{\displaystyle Shi(z)\approx \left({\frac {33317056220720070437}{9686419676455776844590000}}\,{z}^{7}+{\frac {67177799936189717}{98024149196718942600}}\,{z}^{5}+{\frac {540705278447237}{16111793096107650}}\,{z}^{3}+z\right)\left(1-{\frac {177197169001594}{8055896548053825}}\,{z}^{2}+{\frac {87368534024947}{363052404432292380}}\,{z}^{4}-{\frac {212787117226481}{131788022808922133940}}\,{z}^{6}+{\frac {10065927082366801}{1707972775603630855862400}}\,{z}^{8}\right)^{-1}}
Shi(x) Re complex 3D plot Shi(x) Im complex 3D plot Shi(x) abs complex 3D plot
Shi(x) abs complex density plot Shi(x) Re complex density plot Shi(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)
_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)