Tanhc函數定義如下[1]
Tanhc 2D plot
Tanhc'(z) 2D
Tanhc 積分圖
Tanhc integral 3D plot
![{\displaystyle tanhc(z)={\frac {\tanh \left(z\right)}{z}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8695277ecc06fc5ea1e82c459a305d1747f8d217)
- 復域虛部
![{\displaystyle {\it {Im}}\left({\frac {\tanh \left(x+iy\right)}{x+iy}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51f3cbb3959c77b29f0f4a88043b1b689748f519)
- 復域實部
![{\displaystyle {\it {Re}}\left({\frac {\tanh \left(x+iy\right)}{x+iy}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f18858b768419a444a98a18abf2290a6eeafcf5)
- 復域絕對值
![{\displaystyle \left|{\frac {\tanh \left(x+iy\right)}{x+iy}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c9fc7d0456968d5469c6d63175de530c295b021)
- 一階微商
![{\displaystyle {\frac {1-\left(\tanh \left(z\right)\right)^{2}}{z}}-{\frac {\tanh \left(z\right)}{{z}^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a3c35f1d49303b475ac81893af709816e668c75)
- 微商實部
![{\displaystyle -{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e14df201262ba35974878de4b977f9365c3f5787)
- 微商虛部
![{\displaystyle -{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4ed1cc2cdd91befcde21b190b7c9977bc6ad50f)
- 微商絕對值
![{\displaystyle \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|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c803a676306e681b58b11082a4506962bc49cba)
- 積分函數
用其他特殊函數表示[編輯]
![{\displaystyle 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 }}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc1d58f76c253eef1a3b08212182818754adb7a1)
![{\displaystyle 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 }}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/295ab90ffd869058a4a341356b977f2c75889ff1)
![{\displaystyle 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c51d02a3845e99d573b7125df3c9a662bb46d4c)
![{\displaystyle 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/158ce7c53e9fffb6c18567ebbd13b928312789a1)
級數展開[編輯]
Tanhc abs complex 3D
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Tanhc Im complex 3D plot
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Tanhc Re complex 3D plot
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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 abs plot
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Tanhc Im plot
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Tanhc Re plot
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Tanhc'(z) Im plot
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Tanhc'(z) abs plot
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Tanhc'(z) Re 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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參考資料[編輯]
- ^ Weisstein, Eric W. "Tanhc Function." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/TanhcFunction.html (頁面存檔備份,存於網際網路檔案館)