Tanhc函数

Tanhc函数定义如下 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}} 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}}} 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))} Sinhc 函数 Coshc 函数 Tanc 函数 双曲正弦积分函数 Weisstein, Eric W. "Tanhc Function." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/TanhcFunction.html （页面存档备份，存于互联网档案馆）