雙曲正弦積分函數 定義為[ 1] [ 2]
Shi(x) 2D plot
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{\displaystyle {\it {Shi}}\left(z\right)=\int _{0}^{z}\!{\frac {\sinh \left(t\right)}{t}}{dt}}
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{\displaystyle Shi(z)}
常微分方程 的一個解：
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{\displaystyle 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：
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{\displaystyle w\left(z\right)={\it {\_C1}}+{\it {\_C2}}\,{\it {Shi}}\left(z\right)+{\it {\_C3}}\,{\it {Chi}}\left(z\right)}
Meijer G函數
{\displaystyle }
超幾何函數
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{\displaystyle Shi(z)=z*_{1}F_{2}(1/2;3/2,3/2;(1/4)*z^{2})}
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{\displaystyle {\frac {-1}{2}}\,i{\sqrt {\pi }}G_{1,3}^{1,1}\left(-1/4\,{z}^{2}\,{\Big \vert }\,_{1/2,0,0}^{1}\right)}
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18
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600
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35280
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3265920
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439084800
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80951270400
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{\displaystyle {\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
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33317056220720070437
9686419676455776844590000
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67177799936189717
98024149196718942600
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540705278447237
16111793096107650
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177197169001594
8055896548053825
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87368534024947
363052404432292380
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212787117226481
131788022808922133940
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10065927082366801
1707972775603630855862400
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1
{\displaystyle 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}}
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
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
_2D_plot.png)
_Re_complex_3D_plot.png)
_Im_complex_3D_plot.png)
_abs_complex_3D_plot.png)
_abs_complex_density_plot.JPG)
_Re_complex_density_plot.JPG)
_Im_complex_density_plot.JPG)
