Tanc函数

Tanc 函数 定义如下^[1]

${\displaystyle \operatorname {Tanc} (z)={\frac {\tan(z)}{z}}}$

Tanc 2D plot

Tanc'(z) 2D plot

Tanc integral 2D plot

Tanc integral 3D plot

虚域虚部

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

虚域实部

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

绝对值

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

一阶导数

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

导数实部

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

导数虚部

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

导数绝对值

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

与其他特殊函数的关系

• ${\displaystyle \operatorname {Tanc} (z)={\frac {2\,i{{\rm {KummerM}}\left(1,\,2,\,2\,iz\right)}}{\left(2\,z+\pi \right){{\rm {KummerM}}\left(1,\,2,\,i\left(2\,z+\pi \right)\right)}}}}$

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

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

级数展开

${\displaystyle \operatorname {Tanc} z\approx (1+{\frac {1}{3}}{z}^{2}+{\frac {2}{15}}{z}^{4}+{\frac {17}{315}}{z}^{6}+{\frac {62}{2835}}{z}^{8}+{\frac {1382}{155925}}{z}^{10}+{\frac {21844}{6081075}}{z}^{12}+{\frac {929569}{638512875}}{z}^{14}+O\left({z}^{16}\right))}$

${\displaystyle \int _{0}^{z}\!{\frac {\tan \left(x\right)}{x}}{dx}=(z+{\frac {1}{9}}{z}^{3}+{\frac {2}{75}}{z}^{5}+{\frac {17}{2205}}{z}^{7}+{\frac {62}{25515}}{z}^{9}+{\frac {1382}{1715175}}{z}^{11}+{\frac {21844}{79053975}}{z}^{13}+{\frac {929569}{9577693125}}{z}^{15}+O\left({z}^{17}\right))}$

图集

Tanc abs complex 3D

Tanc Im complex 3D plot

Tanc Re complex 3D plot

Tanc'(z) Im complex 3D plot

Tanc'(z) Re complex 3D plot

Tanc'(z) abs complex 3D plot

Tanc abs plot

Tanc Im plot

Tanc Re plot

Tanc'(z) Im plot

Tanc'(z) abs plot

Tanc'(z) Re plot

Tanc integral abs plot

Tanc integral Im plot

Tanc integral Re plot

Tanc abs complex 3D plot

Tanc Im complex 3D plot

Tanc Re complex 3D plot

参看

• Sinhc函数
• Coshc函数
• Tanhc函数
• 双曲正弦积分函数