Si 函数定义如下[1][2] Si(x)的二维图像 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)} 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)} Remove ads表示为其他特殊函数 Meijer G-函数 {\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)} 超几何函数 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})} Remove ads级数展开 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))} Remove ads帕德近似 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}} Remove ads图集 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 参见 Sinhc 函数 Coshc 函数 Tanc 函数 Tanhc 函数 Chi 函数 参考文献Loading content...Loading related searches...Wikiwand - on Seamless Wikipedia browsing. On steroids.Remove ads