Tanc函数

Tanc 函數 定義如下^[1]

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

Tanc 2D plot

Tanc'(z) 2D plot

Tanc integral 2D plot

Tanc integral 3D plot

虛域虛部

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

虛域實部

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

絕對值

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

一階導數

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

導數實部

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

導數虛部

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

導數絕對值

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

與其他特殊函數的關係

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

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

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

級數展開

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

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

圖集

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

參看

• Sinhc函數
• Coshc函數
• Tanhc函數
• 雙曲正弦積分函數