Shi函数

双曲正弦积分函数定义为

{\it Shi} \left( z \right) =\int _{0}^{z}\!{\frac {\sinh \left( t \right) }{t}}{dt}

Shi(z)是下列三阶常微分方程的一个解：

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 \left( z \right) ={\it \_C1}+{\it \_C2}\,{\it Shi} \left( z \right)
+{\it \_C3}\,{\it Chi} \left( z \right)

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

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

级数展开

• {\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
帕德近似
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}

图集
参见
*Sinhc函数
*Coshc函数
*Tanc函数
*Tanhc函数
*Chi函数

参考文献