From Wikipedia, the free encyclopedia
In mathematics, the tanhc function is defined as[1]
![{\displaystyle \operatorname {tanhc} (z)={\frac {\tanh(z)}{z}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c883c1fc8434251fdd0cb64c52a230457ab1ca80)
- Imaginary part in complex plane
![\operatorname {Im}\left({\frac {\tanh(x+iy)}{x+iy}}\right)](https://wikimedia.org/api/rest_v1/media/math/render/svg/9540d970d3785989f5ad8565af0f98363dd3742e)
- Real part in complex plane
![\operatorname {Re}\left({\frac {\tanh \left(x+iy\right)}{x+iy}}\right)](https://wikimedia.org/api/rest_v1/media/math/render/svg/383248912d1507afeeb10de93384dd23f83dc72f)
- absolute magnitude
![\left|{\frac {\tanh(x+iy)}{x+iy}}\right|](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5d5f8f074f7184b6fa0fdb4b35d8953be89e38e)
- First-order derivative
![{\displaystyle {\frac {1-(\tanh(z))^{2}}{z}}-{\frac {\tanh(z)}{z^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa8cb0f1a3c73428026abaa8f7c31d77fcf8b61c)
- Real part of derivative
![{\displaystyle -\operatorname {Re} \left(-{\frac {1-(\tanh(x+iy))^{2}}{x+iy}}+{\frac {\tanh(x+iy)}{(x+iy)^{2}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de4df1998eb929fc0ee3deea655a5d93d43fd877)
- Imaginary part of derivative
![{\displaystyle -\operatorname {Im} \left(-{\frac {1-(\tanh(x+iy))^{2}}{x+iy}}+{\frac {\tanh(x+iy)}{(x+iy)^{2}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d93e7a3b8f357e6cb75b528e8ff812e8b7fe067b)
- absolute value of derivative
![\left|-{\frac {1-(\tanh(x+iy))^{2}}{x+iy}}+{\frac {\tanh(x+iy)}{(x+iy)^{2}}}\right|](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2902ebab3c37c52bbec026b7eb3a766edbe071e)
In terms of other special functions[]
![\operatorname {tanhc}(z)=2\,{{\frac {{{{\rm {KummerM}}}\left(1,\,2,\,2\,z\right)}}{(2\,iz+\pi ){{\rm {KummerM}}}(1,\,2,\,i\pi -2\,z)e^{{2\,z-1/2\,i\pi }}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f4f1e7aa00bebb82ae5d501dc090d6a3624ece5)
![\operatorname {tanhc}(z)=2{\frac {\operatorname {HeunB}(2,0,0,0,{\sqrt {2}}{\sqrt {z}})}{(2iz+\pi )\operatorname {HeunB}(2,0,0,0,{\sqrt {2}}{\sqrt {1/2\,i\pi -z}})e^{{2\,z-1/2\,i\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37ab65705f92f092fa44065c8a907c337db11b63)
![\operatorname {tanhc}(z)={\frac {i{{\rm {\ WhittakerM}}}(0,\,1/2,\,2\,z)}{{{\rm {WhittakerM}}}(0,\,1/2,\,i\pi -2\,z)}}z](https://wikimedia.org/api/rest_v1/media/math/render/svg/a185adb014e89f1fe308895c238ed82015227bb2)
Series expansion[]
![{\displaystyle \operatorname {tanhc} z\approx \left(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(z^{16})\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9725f13077b14b3b0efcb25535f6b560ce224d07)
![{\displaystyle \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))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d73df8c8c06f0058637f2ae0b9e42f6f4668dd1)
Padé approximation[]
![{\displaystyle \operatorname {Tanhc} \left(z\right)=\left(1+{\frac {7}{51}}\,{z}^{2}+{\frac {1}{255}}\,{z}^{4}+{\frac {2}{69615}}\,{z}^{6}+{\frac {1}{34459425}}\,{z}^{8}\right)\left(1+{\frac {8}{17}}\,{z}^{2}+{\frac {7}{255}}\,{z}^{4}+{\frac {4}{9945}}\,{z}^{6}+{\frac {1}{765765}}\,{z}^{8}\right)^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa596214f3c17eab30cb1ef8c49168a42acd9705)
Gallery[]
Tanhc'(z) Im complex 3D plot
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Tanhc'(z) Re complex 3D plot
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Tanhc'(z) abs complex 3D plot
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Tanhc integral abs 3D plot
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Tanhc integral Im 3D plot
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Tanhc integral Re 3D plot
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Tanhc integral abs density plot
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Tanhc integral Im density plot
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Tanhc integral Re density plot
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See also[]
References[]