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#REDIRECT [[Sinc function]]
{{inuse|24H}}
In mathematics, the '''Tanc function''' is defined as<ref>Weisstein, Eric W. "Tanc Function." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/TancFunction.html </ref>
{{Rcat shell|
: <math>\operatorname{Tanc}(z)=\frac {\tan(z) }{z}</math>
{{R to related topic}}
[[File:Tanc 2D plot.png|thumb|Tanc 2D plot]]
[[File:Tanc'(z) 2D plot.png|thumb|Tanc'(z) 2D plot]]
[[File:Tanc integral 2D plot.png|thumb|Tanc integral 2D plot]]
;Imaginary part in complex plane
*<math> \operatorname{Im} \left( \frac {\tan(x+iy) }{x+iy} \right) </math>
;Real part in complex plane
*<math> \operatorname{Re} \left( \frac {\tan \left( x+iy \right) }{x+iy} \right) </math>
;absolute magnitude
*<math> \left| \frac {\tan(x+iy) }{x+iy} \right| </math>
;First-order derivative
*<math> \frac {1- \tan(z))^2}{z} - \frac {\tan(z)}{z^2} </math>
;Real part of derivative
*<math> -\operatorname{Re} \left( -\frac {1- (\tan(x+iy))^2}{x+iy} +\frac{\tan(x+iy)}{(x+iy)^2} \right)
</math>
;Imaginary part of derivative
*<math>-\operatorname{Im} \left( -\frac {1-(\tan(x+iy))^2}{x+iy} + \frac {\tan(x+iy)}{(x+iy)^2} \right)
</math>
;absolute value of derivative
*<math> \left| -\frac{1-(\tan(x+iy))^2}{x+iy}+\frac {\tan(x+iy)}{(x+iy)^2} \right| </math>
==In terms of other special functions==
* <math>\operatorname{Tanc}(z)=2\,{\frac {{{\rm KummerM}\left(1,\,2,\,2\,z\right)}}{(2\,iz+\pi) {\rm KummerM}(1,\,2,\,i\pi -2\,z) e^{2\,z-1/2\,i\pi} }}</math>
*<math>\operatorname{Tanc}(z)=2 \frac {\operatorname{HeunB}(2,0,0,0,\sqrt{2} \sqrt{z})}{( 2iz+\pi) \operatorname{HeunB}( 2,0,0,0,\sqrt{2} \sqrt{1/2\,i\pi -z}) e^{2\,z-1/2\,i\pi}} </math>
* <math>\operatorname{Tanc}(z)= \frac{i{\rm \ WhittakerM}(0,\,1/2,\,2\,z)}{{\rm WhittakerM}(0,\,1/2,\,i\pi -2\,z)} z</math>
==Series expansion==
: <math>\operatorname{Tanc} z \approx \left(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(z^{16}) \right)</math>
==Gallery==
{|
|[[File:Tanc abs complex 3D plot.png|thumb|Tanc abs complex 3D]]
|[[File:Tanc Im complex 3D plot.png|thumb|Tanc Im complex 3D plot]]
|[[File:Tanc Re complex 3D plot.png|thumb|Tanc Re complex 3D plot]]
{|
|[[File:Tanc'(z) Im complex 3D plot.png|thumb|Tanc'(z) Im complex 3D plot]]
|[[File:Tanc'(z) Re complex 3D plot.png|thumb|Tanc'(z) Re complex 3D plot]]
|[[File:Tanc'(z) abs complex 3D plot.png|thumb|Tanc'(z) abs complex 3D plot]]
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|}
{|
|[[File:Tanc abs plot.JPG|thumb|Tanc abs plot]]
|[[File:Tanc Im plot.JPG|thumb|Tanc Im plot]]
|[[File:Tanc Re plot.JPG|thumb|Tanc Re plot]]
|}
{|
|[[File:Tanc'(z) Im plot.JPG|thumb|Tanc'(z) Im plot]]
|[[File:Tanc'(z) abs plot.JPG|thumb|Tanc'(z) abs plot]]
|[[File:Tanc'(z) Re plot.JPG|thumb|Tanc'(z) Re plot]]
|}
==References==
<references/>
[[Category:special functions]]
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