Si 函數 定義如下[ 1] [ 2]
Si(x)的二維圖像
S
i
(
z
)
=
∫
0
z
sin
(
t
)
t
d
t
{\displaystyle {\it {Si}}\left(z\right)=\int _{0}^{z}\!{\frac {\sin \left(t\right)}{t}}{dt}}
S
i
(
z
)
{\displaystyle Si(z)}
S
i
(
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 {\it {Si}}\left(z\right)=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
i
(
z
)
+
_
C
3
C
i
(
z
)
{\displaystyle w\left(z\right)={\it {\_C1}}+{\it {\_C2}}\,{\it {Si}}\left(z\right)+{\it {\_C3}}\,{\it {Ci}}\left(z\right)}
Meijer G函數
{\displaystyle }
−
1
2
π
G
1
,
3
1
,
1
(
1
/
4
z
2
|
1
/
2
,
0
,
0
1
)
{\displaystyle {\frac {-1}{2}}\,{\sqrt {\pi }}G_{1,3}^{1,1}\left(1/4\,{z}^{2}\,{\Big \vert }\,_{1/2,0,0}^{1}\right)}
超幾何函數
S
i
(
z
)
=
z
∗
1
F
2
(
1
/
2
;
3
/
2
,
3
/
2
;
−
(
1
/
4
)
∗
z
2
)
{\displaystyle Si(z)=z*_{1}F_{2}(1/2;3/2,3/2;-(1/4)*z^{2})}
S
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 {Si}}\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))}
S
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 Si(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}}
Si(x) Re complex 3D plot Si(x) Im complex 3D plot Si(x) abs complex 3D plot
Si(x) abs complex density plot Si(x) Re complex density plot Si(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)
