Tanc function

In mathematics, the tanc function is defined for [math]\displaystyle{ z \neq 0 }[/math] as [math]\displaystyle{ \operatorname{tanc}(z)=\frac {\tan(z) }{z} }[/math]

Tanc 2D plot

Tanc'(z) 2D plot

Tanc integral 2D plot

Tanc integral 3D plot

Properties

The first-order derivative of the tanc function is given by

[math]\displaystyle{ \frac{\sec^2(z)}{z} - \frac{\tan(z)}{z^2} }[/math]

The Taylor series expansion is[math]\displaystyle{ \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]which leads to the series expansion of the integral as[math]\displaystyle{ \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]The Padé approximant is[math]\displaystyle{ \operatorname{tanc} \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]

In terms of other special functions

• [math]\displaystyle{ \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], where [math]\displaystyle{ {\rm{KummerM}}(a,b,z) }[/math] is Kummer's confluent hypergeometric function.
• [math]\displaystyle{ \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], where [math]\displaystyle{ {\rm{HeunB}}(q, \alpha, \gamma, \delta, \epsilon ,z) }[/math] is the biconfluent Heun function.
• [math]\displaystyle{ \operatorname{tanc}(z)= \frac {{\rm WhittakerM}(0,\,1/2,\,2\,iz)}{{\rm WhittakerM}(0,\,1/2,\,i (2z+\pi)) z} }[/math], where [math]\displaystyle{ {\rm{WhittakerM}}(a,b,z) }[/math] is a Whittaker function.

Gallery

Tanc abs complex 3D

Tanc Im complex 3D plot

Tanc Re complex 3D plot

See also

• Sinhc function
• Tanhc function
• Coshc function