Tanc 函数 定义如下[1]
![{\displaystyle \operatorname {Tanc} (z)={\frac {\tan(z)}{z}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5de4aa754b088af2cd8d320e81fb913d352eb9f)
Tanc 2D plot
Tanc'(z) 2D plot
Tanc integral 2D plot
Tanc integral 3D plot
- 虚域虚部
![{\displaystyle \operatorname {Im} \left({\frac {\tan(x+iy)}{x+iy}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75c8d40715a635029c88135e3a27d289893078e5)
- 虚域实部
![{\displaystyle \operatorname {Re} \left({\frac {\tan \left(x+iy\right)}{x+iy}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/279499cd7a8556c7aeaf7499c578e52b84df24c5)
- 绝对值
![{\displaystyle \left|{\frac {\tan(x+iy)}{x+iy}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0fdac3cf091f3537d25f49c3e8a1d1aa3e9679bb)
- 一阶导数
![{\displaystyle {\frac {1-\tan(z))^{2}}{z}}-{\frac {\tan(z)}{z^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b394f21bc61750038db5745d800e8131c6a36381)
- 导数实部
![{\displaystyle -\operatorname {Re} \left(-{\frac {1-(\tan(x+iy))^{2}}{x+iy}}+{\frac {\tan(x+iy)}{(x+iy)^{2}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f9615194d830fac7a2e2848b17d2dd4aafd0a78)
- 导数虚部
![{\displaystyle -\operatorname {Im} \left(-{\frac {1-(\tan(x+iy))^{2}}{x+iy}}+{\frac {\tan(x+iy)}{(x+iy)^{2}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b46f64f8b7e488d2dfcb350e682871147ab06fb)
- 导数绝对值
![{\displaystyle \left|-{\frac {1-(\tan(x+iy))^{2}}{x+iy}}+{\frac {\tan(x+iy)}{(x+iy)^{2}}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8b7b61917755482dbeac3f8f8fe494300c719b8)
与其他特殊函数的关系[编辑]
![{\displaystyle \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)}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3fbb70186fcbce46de0f3ffd3eabf27317c7a1cf)
![{\displaystyle \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)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c62bd528184326f50980880e16029da151b1e734)
![{\displaystyle \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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/61f19c493cd79b91a6ef7f291404132a64b21952)
级数展开[编辑]
![{\displaystyle \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))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1bbff41808793eddb992589636e93e4cc8f62d71)
Tanc abs complex 3D
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Tanc Im complex 3D plot
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Tanc Re complex 3D plot
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Tanc'(z) Im complex 3D plot
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Tanc'(z) Re complex 3D plot
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Tanc'(z) abs complex 3D plot
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Tanc abs plot
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Tanc Im plot
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Tanc Re plot
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Tanc'(z) Im plot
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Tanc'(z) abs plot
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Tanc'(z) Re plot
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Tanc integral abs plot
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Tanc integral Im plot
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Tanc integral Re plot
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Tanc abs complex 3D plot
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Tanc Im complex 3D plot
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Tanc Re complex 3D plot
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参考文献[编辑]
- ^ Weisstein, Eric W. "Tanc Function." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/TancFunction.html (页面存档备份,存于互联网档案馆)