双曲正弦积分函数 定义为[1] [2]
Shi(x) 2D plot S h i ( z ) = ∫ 0 z sinh ( t ) t d t {\displaystyle {\it {Shi}}\left(z\right)=\int _{0}^{z}\!{\frac {\sinh \left(t\right)}{t}}{dt}}
S h i ( z ) {\displaystyle Shi(z)} 是下列三阶常微分方程 的一个解:
z d d z w ( z ) − 2 d 2 d z 2 w ( z ) − z d 3 d z 3 w ( z ) = 0 {\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:
w ( z ) = _ C 1 + _ C 2 S h i ( z ) + _ C 3 C h i ( z ) {\displaystyle 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
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Meijer G函数
{\displaystyle } 超几何函数
S h i ( z ) = z ∗ 1 F 2 ( 1 / 2 ; 3 / 2 , 3 / 2 ; ( 1 / 4 ) ∗ z 2 ) {\displaystyle Shi(z)=z*_{1}F_{2}(1/2;3/2,3/2;(1/4)*z^{2})} − 1 2 i π G 1 , 3 1 , 1 ( − 1 / 4 z 2 | 1 / 2 , 0 , 0 1 ) {\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)} 级数展开
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S h i ( z ) = ( z + 1 18 z 3 + 1 600 z 5 + 1 35280 z 7 + 1 3265920 z 9 + 1 439084800 z 11 + 1 80951270400 z 13 + O ( z 15 ) ) {\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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帕德近似
S h i ( z ) ≈ ( 33317056220720070437 9686419676455776844590000 z 7 + 67177799936189717 98024149196718942600 z 5 + 540705278447237 16111793096107650 z 3 + z ) ( 1 − 177197169001594 8055896548053825 z 2 + 87368534024947 363052404432292380 z 4 − 212787117226481 131788022808922133940 z 6 + 10065927082366801 1707972775603630855862400 z 8 ) − 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
参考文献
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^ 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