Tanhc函数定义如下[1] Tanhc 2D plot Tanhc'(z) 2D Tanhc 积分图 Tanhc integral 3D plot t a n h c ( z ) = tanh ( z ) z {\displaystyle tanhc(z)={\frac {\tanh \left(z\right)}{z}}} 复域虚部 I m ( tanh ( x + i y ) x + i y ) {\displaystyle {\it {Im}}\left({\frac {\tanh \left(x+iy\right)}{x+iy}}\right)} 复域实部 R e ( tanh ( x + i y ) x + i y ) {\displaystyle {\it {Re}}\left({\frac {\tanh \left(x+iy\right)}{x+iy}}\right)} 复域绝对值 | tanh ( x + i y ) x + i y | {\displaystyle \left|{\frac {\tanh \left(x+iy\right)}{x+iy}}\right|} 一阶微商 1 − ( tanh ( z ) ) 2 z − tanh ( z ) z 2 {\displaystyle {\frac {1-\left(\tanh \left(z\right)\right)^{2}}{z}}-{\frac {\tanh \left(z\right)}{{z}^{2}}}} 微商实部 − R e ( − 1 − ( tanh ( x + i y ) ) 2 x + i y + tanh ( x + i y ) ( x + i y ) 2 ) {\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)} 微商虚部 − I m ( − 1 − ( tanh ( x + i y ) ) 2 x + i y + tanh ( x + i y ) ( x + i y ) 2 ) {\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)} 微商绝对值 | − 1 − ( tanh ( x + i y ) ) 2 x + i y + tanh ( x + i y ) ( x + i y ) 2 | {\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|} 积分函数 ∫ 0 z tanh ( x ) x d x {\displaystyle \int _{0}^{z}\!{\frac {\tanh \left(x\right)}{x}}{dx}} 用其他特殊函数表示 t a n h c ( z ) = 2 K u m m e r M ( 1 , 2 , 2 z ) ( 2 i z + π ) K u m m e r M ( 1 , 2 , i π − 2 z ) e 2 z − 1 / 2 i π {\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 }}}}} t a n h c ( z ) = 2 H e u n B ( 2 , 0 , 0 , 0 , 2 z ) ( 2 i z + π ) H e u n B ( 2 , 0 , 0 , 0 , 2 1 / 2 i π − z ) e 2 z − 1 / 2 i π {\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 }}}}} t a n h c ( z ) = i W h i t t a k e r M ( 0 , 1 / 2 , 2 z ) W h i t t a k e r M ( 0 , 1 / 2 , i π − 2 z ) z {\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}}} t a n h c ( z ) = i ( e 2 z − 1 ) ( e i π − 2 z − 1 ) e 2 z − 1 / 2 i π z {\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}}} 级数展开 t a n h c ≈ ( 1 − 1 3 z 2 + 2 15 z 4 − 17 315 z 6 + 62 2835 z 8 − 1382 155925 z 10 + 21844 6081075 z 12 − 929569 638512875 z 14 + O ( z 16 ) ) {\displaystyle 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))} ∫ 0 z tanh ( x ) x d x = ( z − 1 9 z 3 + 2 75 z 5 − 17 2205 z 7 + 62 25515 z 9 − 1382 1715175 z 11 + O ( z 13 ) ) {\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))} 图像 Tanhc abs complex 3D Tanhc Im complex 3D plot Tanhc Re complex 3D plot Tanhc'(z) Im complex 3D plot Tanhc'(z) Re complex 3D plot Tanhc'(z) abs complex 3D plot Tanhc abs plot Tanhc Im plot Tanhc Re plot Tanhc'(z) Im plot Tanhc'(z) abs plot Tanhc'(z) Re plot Tanhc integral abs 3D plot Tanhc integral Im 3D plot Tanhc integral Re 3D plot Tanhc integral abs density plot Tanhc integral Im density plot 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Loading content...Loading related searches...Wikiwand - on Seamless Wikipedia browsing. On steroids.