Si 函数

Si 函数定义如下[1][2]

Si(x)的二维图像

${\displaystyle {\it {Si}}\left(z\right)=\int _{0}^{z}\!{\frac {\sin \left(t\right)}{t}}{dt}}$

${\displaystyle Si(z)}$是下列三階常微分方程的一个解:

${\displaystyle {\it {Si}}\left(z\right)=z{\frac {d}{dz}}w\left(z\right)+2\,{\frac {d^{2}}{d{z}^{2}}}w\left(z\right)+z{\frac {d^{3}}{d{z}^{3}}}w\left(z\right)=0}$

${\displaystyle w\left(z\right)={\it {\_C1}}+{\it {\_C2}}\,{\it {Si}}\left(z\right)+{\it {\_C3}}\,{\it {Ci}}\left(z\right)}$

表示为其他特殊函数

• ${\displaystyle {\frac {-1}{2}}\,{\sqrt {\pi }}G_{1,3}^{1,1}\left(1/4\,{z}^{2}\,{\Big \vert }\,_{1/2,0,0}^{1}\right)}$
• ${\displaystyle Si(z)=z*_{1}F_{2}(1/2;3/2,3/2;-(1/4)*z^{2})}$

级数展开

• ${\displaystyle {\it {Si}}\left(z\right)=(z-{\frac {1}{18}}{z}^{3}+{\frac {1}{600}}{z}^{5}-{\frac {1}{35280}}{z}^{7}+{\frac {1}{3265920}}{z}^{9}-{\frac {1}{439084800}}{z}^{11}+{\frac {1}{80951270400}}{z}^{13}+O\left({z}^{15}\right))}$

帕德近似

${\displaystyle Si(z)\approx \left(-{\frac {33317056220720070437}{9686419676455776844590000}}\,{z}^{7}+{\frac {67177799936189717}{98024149196718942600}}\,{z}^{5}-{\frac {540705278447237}{16111793096107650}}\,{z}^{3}+z\right)\left(1+{\frac {177197169001594}{8055896548053825}}\,{z}^{2}+{\frac {87368534024947}{363052404432292380}}\,{z}^{4}+{\frac {212787117226481}{131788022808922133940}}\,{z}^{6}+{\frac {10065927082366801}{1707972775603630855862400}}\,{z}^{8}\right)^{-1}}$

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

参考文献

1. ^ 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.
2. ^ Sloane, N. J. A. Sequence A061079 in "The On-Line Encyclopedia of Integer Sequences