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#REDIRECT [[Sinc_function#Sinhc]]
In mathematics, the '''Sinhc function''' is defined as<ref>Weisstein, Eric W. "Sinhc Function." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/SinhcFunction.html</ref>
{{Rcat shell|
: <math>\operatorname{Sinhc}(z)=\frac {\sinh(z) }{z}</math>
{{R to related topic}}
[[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]]
;Imaginary part in complex plane
*<math> \operatorname{Im} \left( \frac {\sinh(x+iy) }{x+iy} \right) </math>
;Real part in complex plane
*<math> \operatorname{Re} \left( \frac {\sinh \left( x+iy \right) }{x+iy} \right) </math>
;absolute magnitude
*<math> \left| \frac {\sinh(x+iy) }{x+iy} \right| </math>
;First-order derivative
*<math> \frac {1- \sinh(z))^2}{z} - \frac {\sinh(z)}{z^2} </math>
;Real part of derivative
*<math> -\operatorname{Re} \left( -\frac {1- (\sinh(x+iy))^2}{x+iy} +\frac{\sinh(x+iy)}{(x+iy)^2} \right)
</math>
;Imaginary part of derivative
*<math>-\operatorname{Im} \left( -\frac {1-(\sinh(x+iy))^2}{x+iy} + \frac {\sinh(x+iy)}{(x+iy)^2} \right)
</math>
;absolute value of derivative
*<math> \left| -\frac{1-(\sinh(x+iy))^2}{x+iy}+\frac {\sinh(x+iy)}{(x+iy)^2} \right| </math>
==In terms of other special functions==
* <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>
==Series expansion==
: <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>
==Gallery==
{|
|[[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]]
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|}
{|
|[[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]]
|}
==See also==
[[Tanc function]]
[[Tanhc function]]
==References==
<references/>
[[Category:Special functions]]
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