查看“︁Coshc函数”︁的源代码

'''Coshc函数'''常见于有关[[光学散射]]<ref>PN Den Outer, TM Nieuwenhuizen, A Lagendijk,Location of objects in multiple-scattering media,JOSA A, Vol. 10, Issue 6, pp. 1209-1218 (1993)</ref>、[[海森堡时空]]<ref>T Körpinar ,New characterizations for minimizing energy of biharmonic particles in Heisenberg spacetime - International Journal of Theoretical Physics, 2014 - Springer</ref>和[[双曲几何学]]的[[论文]]中<ref>Nilg¨un S¨onmez,A Trigonometric Proof of the Euler Theorem in Hyperbolic Geometry,International Mathematical Forum, 4, 2009, no. 38, 1877 - 1881</ref>其定义如下：<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. "Coshc Function." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/CoshcFunction.html{{dead link|date=2017年11月 |bot=InternetArchiveBot |fix-attempted=yes }}</ref>

: <math>\operatorname{Coshc}(z)=\frac {\cosh(z) }{z}</math>
它是下列微分方程的一个解：

<math>w \left( z \right) z-2\,{\frac {d}{dz}}w \left( z \right) -z{\frac {d^
{2}}{d{z}^{2}}}w \left( z \right) =0</math>

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

;复域虚部
*<math> \operatorname{Im} \left( \frac {\cosh(x+iy) }{x+iy} \right) </math>
;复域实部
*<math> \operatorname{Re} \left( \frac {\cosh \left( x+iy \right) }{x+iy} \right) </math>
;绝对值
*<math>  \left| \frac {\cosh(x+iy) }{x+iy} \right| </math>
;一阶导数
*<math> \frac {\sinh(z)}{z} - \frac {\cosh(z)}{z^2} </math>
;导数实部
*<math> -\operatorname{Re} \left( -\frac {1- (\cosh(x+iy))^2}{x+iy} +\frac{\cosh(x+iy)}{(x+iy)^2} \right)
    </math>
;导数虚部
*<math>-\operatorname{Im} \left( -\frac {1-(\cosh(x+iy))^2}{x+iy} + \frac {\cosh(x+iy)}{(x+iy)^2} \right) 
     </math>
;导数绝对值
*<math>  \left| -\frac{1-(\cosh(x+iy))^2}{x+iy}+\frac {\cosh(x+iy)}{(x+iy)^2} \right| </math>

==表示为其他特殊函数==

* <math>\operatorname{Coshc}(z)={\frac { \left( iz+1/2\,\pi  \right) 
{{\rm M}\left(1,\,2,\,i\pi -2\,z\right)}}{{{\rm e}^{1/2\,i\pi -z}}z}}
</math>

*<math>\operatorname{Coshc}(z)=\frac{1}{2}\,{\frac { \left( 2\,iz+\pi  \right) {\it HeunB} \left( 2,0,0,0,
\sqrt {2}\sqrt {1/2\,i\pi -z} \right) }{{{\rm e}^{1/2\,i\pi -z}}z}}
 </math>

* <math>\operatorname{Coshc}(z)= {\frac {-i \left( 2\,iz+\pi  \right) 
{{\rm \mathbf WhittakerM}\left(0,\,1/2,\,i\pi -2\,z\right)}}{ \left( 4\,iz+2\,\pi 
 \right) z}}
</math>

==级数展开==

: <math>\operatorname{Coshc} z \approx ({z}^{-1}+{\frac {1}{2}}z+{\frac {1}{24}}{z}^{3}+{\frac {1}{720}}{z}^{5}+{\frac {1}{40320}}{z}^{7}+{\frac {1}{3628800}}{z}^{9}+{\frac {1}{
479001600}}{z}^{11}+{\frac {1}{87178291200}}{z}^{13}+O \left( {z}^{15} \right) )</math>

==图集==
{|
|[[File:Coshc abs complex 3D plot.png|thumb|Coshc abs complex 3D]]
|[[File:Coshc Im complex 3D plot.png|thumb|Coshc Im complex 3D plot]]
|[[File:Coshc Re complex 3D plot.png|thumb|Coshc Re complex 3D plot]]
|}
{|
|[[File:Coshc'(z) Im complex 3D plot.png|thumb|Coshc'(z) Im complex 3D plot]]
|[[File:Coshc'(z) Re complex 3D plot.png|thumb|Coshc'(z) Re complex 3D plot]]
|[[File:Coshc'(z) abs complex 3D plot.png|thumb|Coshc'(z) abs complex 3D plot]]
|
|}
{|
|[[File:Coshc'(x) abs density plot.JPG|thumb|Coshc'(x) abs density plot]]
|[[File:Coshc'(x) Im density plot.JPG|thumb|Coshc'(x) Im density plot]]
|[[File:Coshc'(x) Re density plot.JPG|thumb|Coshc'(x) Re density plot]]
|}

==参见==
*[[Tanc函数]]
*[[Tanhc函数]]
*[[Sinhc函数]]

==参考文献==
<references/>

[[Category:特殊函数]]