查看“︁Shi函数”︁的源代码

'''双曲正弦积分函数'''定义为<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:Shi(x) 2D plot.png|thumb|300px|Shi(x) 2D plot]]
<math>{\it Shi} \left( z \right) =\int _{0}^{z}\!{\frac {\sinh \left( t \right) }{t}}{dt}</math>

<math>Shi(z)</math>是下列三阶[[常微分方程]]的一个解：

<math>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 Shi} \left( z \right) 
+{\it \_C3}\,{\it Chi} \left( z \right)</math>

==与其他特殊函数的关系==
[[Meijer G函数]]
*<math>     </math>
[[超几何函数]]
*<math>Shi(z)=z*_{1}F_{2}(1/2;3/2, 3/2; (1/4)*z^2)     </math>

*<math>\frac{ -1}{2}\,i\sqrt {\pi }
G^{1, 1}_{1, 3}\left(-1/4\,{z}^{2}\, \Big\vert\,^{1}_{1/2, 0, 0}\right)
    </math>

==级数展开==
*<math> {\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) )
    </math>
==帕德近似==
[[帕德近似]]
<math>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}</math>

==图集==
{|
|[[File:Shi(x) Re complex 3D plot.png|thumb|Shi(x) Re complex 3D plot]]
|[[File:Shi(x) Im complex 3D plot.png|thumb|Shi(x) Im complex 3D plot]]
|[[File:Shi(x) abs complex 3D plot.png|thumb|Shi(x) abs complex 3D plot]]
|}
{|
|[[File:Shi(x) abs complex density plot.JPG|thumb|Shi(x) abs complex density plot]]
|[[File:Shi(x) Re complex density plot.JPG|thumb|Shi(x) Re complex density plot]]
|[[File:Shi(x) Im complex density plot.JPG|thumb|Shi(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:特殊函數]]