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)}
與其他特殊函數的關係
編輯
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)}
級數展開
編輯
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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
編輯
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
參考文獻
編輯
^ 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)