Tanc函数

Tanc 函数 定义如下

: \operatorname{Tanc}(z)=\frac {\tan(z) }{z}

;虚域虚部

• \operatorname{Im} \left( \frac {\tan(x+iy) }{x+iy} \right)

;虚域实部

• \operatorname{Re} \left( \frac {\tan \left( x+iy \right) }{x+iy} \right)

;绝对值

• \left| \frac {\tan(x+iy) }{x+iy} \right|

;一阶导数

• \frac {1- \tan(z))^2}{z} - \frac {\tan(z)}{z^2}

;导数实部

• -\operatorname{Re} \left( -\frac {1- (\tan(x+iy))^2}{x+iy} +\frac{\tan(x+iy)}{(x+iy)^2} \right)

;导数虚部
*-\operatorname{Im} \left( -\frac {1-(\tan(x+iy))^2}{x+iy} + \frac {\tan(x+iy)}{(x+iy)^2} \right)

;导数绝对值

• \left| -\frac{1-(\tan(x+iy))^2}{x+iy}+\frac {\tan(x+iy)}{(x+iy)^2} \right|

与其他特殊函数的关系

• \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)}}}

*\operatorname{Tanc}(z)={\frac {2\,i{\it HeunB} \left( 2,0,0,0,\sqrt {2}\sqrt {iz} \right) }{
\left( 2\,z+\pi \right) {\it HeunB} \left( 2,0,0,0,\sqrt {2}\sqrt {1
/2\,i \left( 2\,z+\pi \right) } \right) }}

• \operatorname{Tanc}(z)={\frac {{{\rm WhittakerM}\left(0,\,1/2,\,2\,iz\right)}}{

{{\rm WhittakerM}\left(0,\,1/2,\,i \left( 2\,z+\pi \right) \right)}z}}

级数展开
: \operatorname{Tanc} 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) )

\int _{0}^{z}\!{\frac {\tan \left( x \right) }{x}}{dx}=(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 \left( {z}^{
17} \right) )

图集
参看
*Sinhc函数
*Coshc函数
*Tanhc函数
*双曲正弦积分函数

参考文献