Sinhc函数
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- <math>\operatorname{Sinhc}(z)=\frac {\sinh(z) }{z}</math>，它是下列微分方程的一个解：
- <math>w(z) z-2\,\frac {d}{dz} w (z) -z \frac {d^2}{dz^2} w (z) =0</math>
_2D_plot.png){kind=link}
- 复域虚部
- <math> \operatorname{Im} \left( \frac {\sinh(x+iy) }{x+iy} \right) </math>
- 复域实部
- <math> \operatorname{Re} \left( \frac {\sinh \left( x+iy \right) }{x+iy} \right) </math>
- 绝对值
- <math> \left| \frac {\sinh(x+iy) }{x+iy} \right| </math>
- 一阶导数
- <math> \frac {\cosh(z)}{z} - \frac {\sinh(z)}{z^2} </math>
- 导数实部
- <math> -\operatorname{Re} \left( -\frac {1- (\sinh(x+iy))^2}{x+iy} +\frac{\sinh(x+iy)}{(x+iy)^2} \right)
</math>
- 导数虚部
- <math>-\operatorname{Im} \left( -\frac {1-(\sinh(x+iy))^2}{x+iy} + \frac {\sinh(x+iy)}{(x+iy)^2} \right)
</math>
- 绝对值
- <math> \left| -\frac{1-(\sinh(x+iy))^2}{x+iy}+\frac {\sinh(x+iy)}{(x+iy)^2} \right| </math>
表示为其他特殊函数[编辑]
- <math>\operatorname{Sinhc}(z)={\frac {{{\rm KummerM}\left(1,\,2,\,2\,z\right)}}{{{\rm e}^{z}}}}</math>
- <math>\operatorname{Sinhc}(z)={\frac {{\it HeunB} \left( 2,0,0,0,\sqrt {2}\sqrt {z} \right) }{{
{\rm e}^{z}}}} </math>
- <math>\operatorname{Sinhc}(z)=1/2\,{\frac {{{\rm WhittakerM}\left(0,\,1/2,\,2\,z\right)}}{z}} </math>
级数展开[编辑]
- <math>\operatorname{Sinhc} z \approx (1+{\frac {1}{3}}{z}^{2}+{\frac {2}{15}}{z}^{4}+{\frac {17}{315}}{z}^{6}+{\frac {62}{2835}}{z}^{8}+{\frac {1382}{155925}}{z}^{10}+{\frac {
21844}{6081075}}{z}^{12}+{\frac {929569}{638512875}}{z}^{14}+O \left( {z}^{16} \right) )</math>
图集[编辑]
_Im_complex_3D_plot.png){kind=link}
_Re_complex_3D_plot.png){kind=link}
_abs_complex_3D_plot.png){kind=link}
_Im_plot.JPG){kind=link}
_abs_plot.JPG){kind=link}
_Re_plot.JPG){kind=link}
参见[编辑]
参考文献[编辑]
- ↑ JHM ten Thije Boonkkamp, J van Dijk, L Liu,Extension of the complete flux scheme to systems of conservation laws,J Sci Comput (2012) 53:552–568,DOI 10.1007/s10915-012-9588-5
- ↑ Weisstein, Eric W. "Sinhc Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SinhcFunction.html （页面存档备份，存于互联网档案馆）
- ↑ P. N. den Outer, Th. M. Nieuwenhuizen, and Ad Lagendijk,Location of objects in multiple-scattering media,JOSA A, Vol. 10, Issue 6, pp. 1209-1218 (1993)