Si 函数定义如下
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)} 是下列三阶常微分方程的一个解:
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函数
超几何函数
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}}