Tanc函数

Tanc 函数 定义如下^[1]

\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))

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