查看“︁Si函数”︁的源代码

'''Si 函数'''定义如下<ref>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. </ref><ref>
Sloane, N. J. A. Sequence A061079 in "The On-Line Encyclopedia of Integer Sequences</ref>

[[File:Si(x) 2D plot.png|thumb|300px|Si(x)的二维图像]]
<math>{\it Si} \left( z \right) =\int _{0}^{z}\!{\frac {\sin \left( t \right) }{t}}{dt}</math>

<math>Si(z)</math>是下列三階常微分方程的一个解:

<math>{\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</math>

即：

<math>w \left( z \right) ={\it \_C1}+{\it \_C2}\,{\it Si} \left( z \right) 
+{\it \_C3}\,{\it Ci} \left( z \right)</math>

==表示为其他特殊函数==
[[Meijer G-函数]]
*<math>     </math>
*<math>\frac{ -1}{2}\,\sqrt {\pi }
G^{1, 1}_{1, 3}\left(1/4\,{z}^{2}\, \Big\vert\,^{1}_{1/2, 0, 0}\right)    </math>
[[超几何函数]]

*<math>Si(z)=z*_{1}F_{2}(1/2;3/2, 3/2; -(1/4)*z^2)     </math>

==级数展开==
*<math> {\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) )    </math>

==帕德近似==
<math>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}</math>

==图集==
{|
|[[File:Si(x) Re complex 3D plot.png|thumb|Si(x) Re complex 3D plot]]
|[[File:Si(x) Im complex 3D plot.png|thumb|Si(x) Im complex 3D plot]]
|[[File:Si(x) abs complex 3D plot.png|thumb|Si(x) abs complex 3D plot]]
|}
{|
|[[File:Si(x) abs complex density plot.JPG|thumb|Si(x) abs complex density plot]]
|[[File:Si(x) Re complex density plot.JPG|thumb|Si(x) Re complex density plot]]
|[[File:Si(x) Im complex density plot.JPG|thumb|Si(x) Im complex density plot]]
|
|}
==参见==
*[[Sinhc 函数]]
*[[Coshc 函数]]
*[[Tanc 函数]]
*[[Tanhc 函数]]
*[[Chi 函数]]

==参考文献==
<references/>

[[Category:特殊函数]]