Shi函数

双曲正弦积分函数定义为[1][2]

Shi(x) 2D plot

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

$Shi(z)$ 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

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

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

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

Shi(x) Re complex 3D plot

Shi(x) Im complex 3D plot

Shi(x) abs complex 3D plot

Shi(x) abs complex density plot

Shi(x) Re complex density plot

Shi(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