反双曲函数

反雙曲函數示意圖 反双曲函数是双曲函数的反函数。与反圆函数不同之处是它的前缀是ar意即area（面积），而不是arc（弧）。因为双曲角是以双曲线、通过原点直线以及其对x轴的映射三者之间所夹面积定义的，而圆角是以弧长与半径的比值定义。 符号 s i n h − 1 , c o s h − 1 {\displaystyle \mathrm {sinh} ^{-1},\mathrm {cosh} ^{-1}} 等常用于 a r s i n h , a r c o s h {\displaystyle \mathrm {arsinh} ,\mathrm {arcosh} } 等。但是这种符号有时在 s i n h − 1 x {\displaystyle \mathrm {sinh} ^{-1}x} 和 1 s i n h x {\displaystyle {\frac {1}{\mathrm {sinh} x}}} 之间易造成混淆。 下表列出基本的反双曲函数。 d d x arsinh x = 1 1 + x 2 d d x arcosh x = 1 x 2 − 1 , x > 1 d d x artanh x = 1 1 − x 2 , | x | < 1 d d x arcoth x = 1 1 − x 2 , | x | > 1 d d x arsech x = − 1 x 1 − x 2 , x ∈ ( 0 , 1 ) d d x arcsch x = − 1 | x | 1 + x 2 , x  ≠  0 {\displaystyle {\begin{aligned}{\frac {d}{dx}}\operatorname {arsinh} \,x&{}={\frac {1}{\sqrt {1+x^{2}}}}\\{\frac {d}{dx}}\operatorname {arcosh} \,x&{}={\frac {1}{\sqrt {x^{2}-1}}},\qquad x>1\\{\frac {d}{dx}}\operatorname {artanh} \,x&{}={\frac {1}{1-x^{2}}},\qquad |x|<1\\{\frac {d}{dx}}\operatorname {arcoth} \,x&{}={\frac {1}{1-x^{2}}},\qquad |x|>1\\{\frac {d}{dx}}\operatorname {arsech} \,x&{}={\frac {-1}{x{\sqrt {1-x^{2}}}}},\qquad x\in (0,1)\\{\frac {d}{dx}}\operatorname {arcsch} \,x&{}={\frac {-1}{|x|{\sqrt {1+x^{2}}}}},\qquad x{\text{ ≠ }}0\\\end{aligned}}} 求导范例： 设θ = arsinh x，则： d arsinh x d x = d θ d sinh ⁡ θ = 1 cosh ⁡ θ = 1 1 + sinh 2 ⁡ θ = 1 1 + x 2 {\displaystyle {\frac {d\,\operatorname {arsinh} \,x}{dx}}={\frac {d\theta }{d\sinh \theta }}={\frac {1}{\cosh \theta }}={\frac {1}{\sqrt {1+\sinh ^{2}\theta }}}={\frac {1}{\sqrt {1+x^{2}}}}} arsinh x {\displaystyle \operatorname {arsinh} \,x} = x − ( 1 2 ) x 3 3 + ( 1 ⋅ 3 2 ⋅ 4 ) x 5 5 − ( 1 ⋅ 3 ⋅ 5 2 ⋅ 4 ⋅ 6 ) x 7 7 + ⋯ {\displaystyle =x-\left({\frac {1}{2}}\right){\frac {x^{3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {x^{5}}{5}}-\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {x^{7}}{7}}+\cdots } = ∑ n = 0 ∞ ( ( − 1 ) n ( 2 n ) ! 2 2 n ( n ! ) 2 ) x 2 n + 1 ( 2 n + 1 ) , | x | < 1 {\displaystyle =\sum _{n=0}^{\infty }\left({\frac {(-1)^{n}(2n)!}{2^{2n}(n!)^{2}}}\right){\frac {x^{2n+1}}{(2n+1)}},\qquad \left|x\right|<1} arcosh x {\displaystyle \operatorname {arcosh} \,x} = ln ⁡ 2 x − ( ( 1 2 ) x − 2 2 + ( 1 ⋅ 3 2 ⋅ 4 ) x − 4 4 + ( 1 ⋅ 3 ⋅ 5 2 ⋅ 4 ⋅ 6 ) x − 6 6 + ⋯ ) {\displaystyle =\ln 2x-\left(\left({\frac {1}{2}}\right){\frac {x^{-2}}{2}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {x^{-4}}{4}}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {x^{-6}}{6}}+\cdots \right)} = ln ⁡ 2 x − ∑ n = 1 ∞ ( ( − 1 ) n ( 2 n ) ! 2 2 n ( n ! ) 2 ) x − 2 n ( 2 n ) , x > 1 {\displaystyle =\ln 2x-\sum _{n=1}^{\infty }\left({\frac {(-1)^{n}(2n)!}{2^{2n}(n!)^{2}}}\right){\frac {x^{-2n}}{(2n)}},\qquad x>1} artanh x = x + x 3 3 + x 5 5 + x 7 7 + ⋯ = ∑ n = 0 ∞ x 2 n + 1 ( 2 n + 1 ) , | x | < 1 {\displaystyle \operatorname {artanh} \,x=x+{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}+{\frac {x^{7}}{7}}+\cdots =\sum _{n=0}^{\infty }{\frac {x^{2n+1}}{(2n+1)}},\qquad \left|x\right|<1} arcsch x = arsinh x − 1 {\displaystyle \operatorname {arcsch} \,x=\operatorname {arsinh} \,x^{-1}} = x − 1 − ( 1 2 ) x − 3 3 + ( 1 ⋅ 3 2 ⋅ 4 ) x − 5 5 − ( 1 ⋅ 3 ⋅ 5 2 ⋅ 4 ⋅ 6 ) x − 7 7 + ⋯ {\displaystyle =x^{-1}-\left({\frac {1}{2}}\right){\frac {x^{-3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {x^{-5}}{5}}-\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {x^{-7}}{7}}+\cdots } = ∑ n = 0 ∞ ( ( − 1 ) n ( 2 n ) ! 2 2 n ( n ! ) 2 ) x − ( 2 n + 1 ) ( 2 n + 1 ) , | x | < 1 {\displaystyle =\sum _{n=0}^{\infty }\left({\frac {(-1)^{n}(2n)!}{2^{2n}(n!)^{2}}}\right){\frac {x^{-(2n+1)}}{(2n+1)}},\qquad \left|x\right|<1} arsech x = arcosh x − 1 {\displaystyle \operatorname {arsech} \,x=\operatorname {arcosh} \,x^{-1}} = ln ⁡ 2 x − ( ( 1 2 ) x 2 2 + ( 1 ⋅ 3 2 ⋅ 4 ) x 4 4 + ( 1 ⋅ 3 ⋅ 5 2 ⋅ 4 ⋅ 6 ) x 6 6 + ⋯ ) {\displaystyle =\ln {\frac {2}{x}}-\left(\left({\frac {1}{2}}\right){\frac {x^{2}}{2}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {x^{4}}{4}}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {x^{6}}{6}}+\cdots \right)} = ln ⁡ 2 x − ∑ n = 1 ∞ ( ( − 1 ) n ( 2 n ) ! 2 2 n ( n ! ) 2 ) x 2 n 2 n , 0 < x ≤ 1 {\displaystyle =\ln {\frac {2}{x}}-\sum _{n=1}^{\infty }\left({\frac {(-1)^{n}(2n)!}{2^{2n}(n!)^{2}}}\right){\frac {x^{2n}}{2n}},\qquad 0<x\leq 1} arcoth x = artanh x − 1 {\displaystyle \operatorname {arcoth} \,x=\operatorname {artanh} \,x^{-1}} = x − 1 + x − 3 3 + x − 5 5 + x − 7 7 + ⋯ {\displaystyle =x^{-1}+{\frac {x^{-3}}{3}}+{\frac {x^{-5}}{5}}+{\frac {x^{-7}}{7}}+\cdots } = ∑ n = 0 ∞ x − ( 2 n + 1 ) ( 2 n + 1 ) , | x | > 1 {\displaystyle =\sum _{n=0}^{\infty }{\frac {x^{-(2n+1)}}{(2n+1)}},\qquad \left|x\right|>1} arcosh ⁡ ( 2 x 2 − 1 ) = 2 arcosh ⁡ x {\displaystyle \operatorname {arcosh} (2x^{2}-1)=2\operatorname {arcosh} x} arcosh ⁡ ( 2 x 2 + 1 ) = 2 arsinh ⁡ x {\displaystyle \operatorname {arcosh} (2x^{2}+1)=2\operatorname {arsinh} x} ∫ arsinh x d x = x arsinh x − x 2 + 1 + C ∫ arcosh x d x = x arcosh x − x 2 − 1 + C , x > 1 ∫ artanh x d x = x artanh x + 1 2 ln ⁡ ( 1 − x 2 ) + C , | x | < 1 ∫ arcoth x d x = x arcoth x + 1 2 ln ⁡ ( x 2 − 1 ) + C , | x | > 1 ∫ arsech x d x = x arsech x + arcsin x + C , x ∈ ( 0 , 1 ) ∫ arcsch x d x = x arcsch x + | arsinh x | + C , x ≠ 0 {\displaystyle {\begin{aligned}\int \operatorname {arsinh} \,x\,dx&{}=x\,\operatorname {arsinh} \,x-{\sqrt {x^{2}+1}}+C\\\int \operatorname {arcosh} \,x\,dx&{}=x\,\operatorname {arcosh} \,x-{\sqrt {x^{2}-1}}+C,\qquad x>1\\\int \operatorname {artanh} \,x\,dx&{}=x\,\operatorname {artanh} \,x+{\frac {1}{2}}\ln \left(1-x^{2}\right)+C,\qquad |x|<1\\\int \operatorname {arcoth} \,x\,dx&{}=x\,\operatorname {arcoth} \,x+{\frac {1}{2}}\ln \left(x^{2}-1\right)+C,\qquad |x|>1\\\int \operatorname {arsech} \,x\,dx&{}=x\,\operatorname {arsech} \,x+\arcsin \,x+C,\qquad x\in (0,1)\\\int \operatorname {arcsch} \,x\,dx&{}=x\,\operatorname {arcsch} \,x+\left|\operatorname {arsinh} \,x\right|+C,\qquad x\neq 0\end{aligned}}} 使用分部积分法和上面的简单导数很容易得出它们。 双曲余弦函数是偶函数，所以对于一个y值（y>1），都有两个x值与之对应，取反的时候只取一个（通常是正的）即可。 Inverse trigonometric functions（页面存档备份，存于互联网档案馆） at MathWorld 双曲函数