查看“Sinhc函数”的源代码

数学上的'''Sinhc函数'''定义如下<ref>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</ref><ref>Weisstein, Eric W. "Sinhc Function." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/SinhcFunction.html </ref><ref>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)</ref>

: <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>

[[File:Sinhc 2D plot.png|thumb|Sinhc 2D plot]]
[[File:Sinhc'(z) 2D plot.png|thumb|Sinhc'(z) 2D plot]]
[[File:Sinhc integral 2D plot.png|thumb|Sinhc integral 2D plot]]

;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 {1- \sinh(z))^2}{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==

* <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==

: <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==
{|
|[[File:Sinhc abs complex 3D plot.png|thumb|Sinhc abs complex 3D]]
|[[File:Sinhc Im complex 3D plot.png|thumb|Sinhc Im complex 3D plot]]
|[[File:Sinhc Re complex 3D plot.png|thumb|Sinhc Re complex 3D plot]]
|}
{|
|[[File:Sinhc'(z) Im complex 3D plot.png|thumb|Sinhc'(z) Im complex 3D plot]]
|[[File:Sinhc'(z) Re complex 3D plot.png|thumb|Sinhc'(z) Re complex 3D plot]]
|[[File:Sinhc'(z) abs complex 3D plot.png|thumb|Sinhc'(z) abs complex 3D plot]]
|
|}

{|
|[[File:Sinhc abs plot.JPG|thumb|Sinhc abs plot]]  
|[[File:Sinhc Im plot.JPG|thumb|Sinhc Im plot]]
|[[File:Sinhc Re plot.JPG|thumb|Sinhc Re plot]]
|}
{|
|[[File:Sinhc'(z) Im plot.JPG|thumb|Sinhc'(z) Im plot]]
|[[File:Sinhc'(z) abs plot.JPG|thumb|Sinhc'(z) abs plot]]
|[[File:Sinhc'(z) Re plot.JPG|thumb|Sinhc'(z) Re plot]]
|}
==参见==
*[[Tanhc函数]]
*[[Coshc函数]]
*[[Tanc函数]]
*[[双曲正弦积分函数]]

==参考文献==
<references/>

[[Category:特殊函数]]