*#REDIRECT [[ SinhcSinc function]] ▼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.png|thumb|Tanc integral 2D plot]]
[[File:Tanc integral 3D plot.png|thumb|Tanc integral 3D 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)={\frac {2\,i{{\rm KummerM}\left(1,\,2,\,2\,iz\right)}}{ \left( 2\,z+\pi
\right) {{\rm KummerM}\left(1,\,2,\,i \left( 2\,z+\pi \right) \right)}}}</math>
*<math>\operatorname{Tanc}(z)= \frac {2i \operatorname{HeunB} \left( 2,0,0,0,\sqrt {2}\sqrt {iz} \right) }{(2z+\pi) \operatorname{HeunB} \left( 2,0,0,0,\sqrt {2}\sqrt {(i/2) (2z+\pi) } \right) } </math>
* <math>\operatorname{Tanc}(z)= \frac {{\rm WhittakerM}(0,\,1/2,\,2\,iz)}{{\rm WhittakerM}(0,\,1/2,\,i (2z+\pi)) 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>
: <math>\int _0^z \frac {\tan(x) }{x} \, dx = \left(z+ \frac {1}{9} z^3 + \frac {2}{75} z^5 + \frac {17}{2205} z^7 + \frac {62}{25515} z^9+ \frac {1382}{1715175} z^{11}+ \frac {21844}{
79053975} z^{13} + \frac{929569}{9577693125} z^{15}+ O (z^{17}) \right)</math>
==Pade approximation==
<math>{\it Tainc} \left( z \right) = \left( 1-{\frac {7}{51}}\,{z}^{2}+{
\frac {1}{255}}\,{z}^{4}-{\frac {2}{69615}}\,{z}^{6}+{\frac {1}{
34459425}}\,{z}^{8} \right) \left( 1-{\frac {8}{17}}\,{z}^{2}+{\frac
{7}{255}}\,{z}^{4}-{\frac {4}{9945}}\,{z}^{6}+{\frac {1}{765765}}\,{z}
^{8} \right) ^{-1}
</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]]
|
|}
{|
|[[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]]
|}
{|
|[[File:Tanc integral abs plot.png|thumb|Tanc integral abs plot]]
|[[File:Tanc integral Im plot.png|thumb|Tanc integral Im plot]]
|[[File:Tanc integral Re plot.png|thumb|Tanc integral Re plot]]
|}
{|
|[[File:Tanc abs complex 3D plot2.JPG|thumb|Tanc abs complex 3D plot]]
|[[File:Tanc Im complex 3D plot2.JPG|thumb|Tanc Im complex 3D plot]]
|[[File:Tanc Re complex 3D plot2.JPG|thumb|Tanc Re complex 3D plot]]
|}
==See also==
* [[Tanhc function]]
* [[Coshc function]]
==References==
<references/>
[[Category:Special functions]]
| 