查看“Tanhc函数”的源代码

'''Tanhc'''函数定义如下<ref>Weisstein, Eric W. "Tanhc Function." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/TanhcFunction.html </ref>
[[File:Tanhc 2D plot.png|thumb|Tanhc 2D plot]]
[[File:Tanhc'(z) 2D plot.png|thumb|Tanhc'(z) 2D ]]
[[File:Tanhc integral 2D plot.png|thumb|Tanhc 积分图]]
[[File:Tanhc integral 3D plot.png|thumb|Tanhc integral 3D plot]]

*<math> tanhc(z)={\frac {\tanh \left( z \right) }{z}}    </math>
;复域虚部
*<math> {\it Im} \left( {\frac {\tanh \left( x+iy \right) }{x+iy}} \right)     </math>
;复域实部
*<math>{\it Re} \left( {\frac {\tanh \left( x+iy \right) }{x+iy}} \right)      </math>
;复域绝对值
*<math>  \left| {\frac {\tanh \left( x+iy \right) }{x+iy}} \right|     </math>
;一阶微商
*<math>{\frac {1- \left( \tanh \left( z \right)  \right) ^{2}}{z}}-{\frac {
\tanh \left( z \right) }{{z}^{2}}}
     </math>
;微商实部
*<math> -{\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)
    </math>
;微商虚部
*<math>-{\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) 
     </math>
;微商绝对值
*<math>  \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| 
    </math>
;积分函数
<math>\int _{0}^{z}\!{\frac {\tanh \left( x \right) }{x}}{dx}</math>

==用其他特殊函数表示==
*<math>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 }}}}</math>
*<math>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 }}}}</math>
*<math>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}}</math>
*<math>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}}</math>

==级数展开==
<math>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) )</math>

<math>\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) )</math>

==图像==
{|
|[[File:Tanhc abs complex 3D plot.png|thumb|Tanhc abs complex 3D]]
|[[File:Tanhc Im complex 3D plot.png|thumb|Tanhc Im complex 3D plot]]
|[[File:Tanhc Re complex 3D plot.png|thumb|Tanhc Re complex 3D plot]]
|}
{|
|[[File:Tanhc'(z) Im complex 3D plot.png|thumb|Tanhc'(z) Im complex 3D plot]]
|[[File:Tanhc'(z) Re complex 3D plot.png|thumb|Tanhc'(z) Re complex 3D plot]]
|[[File:Tanhc'(z) abs complex 3D plot.png|thumb|Tanhc'(z) abs complex 3D plot]]
|
|}

{|
|[[File:Tanhc abs plot.JPG|thumb|Tanhc abs plot]]  
|[[File:Tanhc Im plot.JPG|thumb|Tanhc Im plot]]
|[[File:Tanhc Re plot.JPG|thumb|Tanhc Re plot]]
|}
{|
|[[File:Tanhc'(z) Im plot.JPG|thumb|Tanhc'(z) Im plot]]
|[[File:Tanhc'(z) abs plot.JPG|thumb|Tanhc'(z) abs plot]]
|[[File:Tanhc'(z) Re plot.JPG|thumb|Tanhc'(z) Re plot]]
|}
{|
|[[File:Tanhc integral abs 3D plot.png|thumb|Tanhc integral abs 3D plot]]
|[[File:Tanhc integral Im 3D plot.png|thumb|Tanhc integral Im 3D plot]]
|[[File:Tanhc integral Re 3D plot.png|thumb|Tanhc integral Re 3D plot]]
|}
{|
|[[File:Tanhc integral abs density plot.JPG|thumb|Tanhc integral abs density plot]]
|[[File:Tanhc integral Im density plot.JPG|thumb|Tanhc integral Im density plot]]
|[[File:Tanhc integral Re density plot.JPG|thumb|Tanhc integral Re density plot]]
|}

==参看==
*[[Sinhc 函数]]
*[[Coshc 函数]]
*[[Tanc 函数]]
*[[双曲正弦积分函数]]

==参考资料==
<references/>

[[Category:特殊函数]]