Tanhc函数

Tanhc函数定义如下

• tanhc(z)={\frac {\tanh \left( z \right) }{z}}

;复域虚部

• {\it Im} \left( {\frac {\tanh \left( x+iy \right) }{x+iy}} \right)

;复域实部
*{\it Re} \left( {\frac {\tanh \left( x+iy \right) }{x+iy}} \right)
;复域绝对值

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

;一阶微商
*{\frac {1- \left( \tanh \left( z \right) \right) ^{2}}{z}}-{\frac {
\tanh \left( z \right) }{{z}^{2}}}

;微商实部

• -{\it Re} \left( -{\frac {1- \left( \tanh \left( x+iy \right)

\right) ^{2}}{x+iy}}+{\frac {\tanh \left( x+iy \right) }{ \left( x+iy
\right) ^{2}}} \right)

;微商虚部
*-{\it Im} \left( -{\frac {1- \left( \tanh \left( x+iy \right)
\right) ^{2}}{x+iy}}+{\frac {\tanh \left( x+iy \right) }{ \left( x+iy
\right) ^{2}}} \right)

;微商绝对值

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

iy}}+{\frac {\tanh \left( x+iy \right) }{ \left( x+iy \right) ^{2}}}
\right|

;积分函数
\int _{0}^{z}\!{\frac {\tanh \left( x \right) }{x}}{dx}

用其他特殊函数表示
*tanhc(z)=2\,{\frac {{{\rm KummerM}\left(1,\,2,\,2\,z\right)}}{ \left( 2\,iz+\pi
\right) {{\rm KummerM}\left(1,\,2,\,i\pi -2\,z\right)}{{\rm e}^{2\,z-1/2\,i\pi }}}}
*tanhc(z)=2\,{\frac {{\it HeunB} \left( 2,0,0,0,\sqrt {2}\sqrt {z} \right) }{
\left( 2\,iz+\pi \right) {\it HeunB} \left( 2,0,0,0,\sqrt {2}\sqrt {
1/2\,i\pi -z} \right) {{\rm e}^{2\,z-1/2\,i\pi }}}}
*tanhc(z)={\frac {i{{\rm \ WhittakerM}\left(0,\,1/2,\,2\,z\right)}}{
{{\rm WhittakerM}\left(0,\,1/2,\,i\pi -2\,z\right)}z}}
*tanhc(z)={\frac {i \left( {{\rm e}^{2\,z}}-1 \right) }{ \left( {{\rm e}^{i\pi -
2\,z}}-1 \right) {{\rm e}^{2\,z-1/2\,i\pi }}z}}

级数展开
tanhc \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 {\tanh \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}+O \left( {z}^{13}
\right) )

图像
参看
*Sinhc 函数
*Coshc 函数
*Tanc 函数
*双曲正弦积分函数

参考资料