Si函数

Si 函数定义如下[1][2]

Si(x)的二维图像

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

$Si(z)$ 是下列三階常微分方程的一个解:

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

即：

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

表示为其他特殊函数

Meijer G函数

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

超几何函数

$Si(z)=z*_{1}F_{2}(1/2;3/2,3/2;-(1/4)*z^{2})$

级数展开

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

帕德近似

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

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

参见

参考文献

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

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[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