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。
数学符号
符号
等常用于
等。但是这种符号有时在
和
之间易造成混淆。
主值
下表列出基本的反双曲函数。
名称
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常用符号
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定义
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定义域
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值域
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图像
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反双曲正弦 |
![{\displaystyle y=\mathrm {arsinh} x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db9877835dee4645965cb4be96060437005abb0e) |
![{\displaystyle \ln(x+{\sqrt {x^{2}+1}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/274d0dac7bde372b9eedd93c57e2bdbc0b5424aa) |
![{\displaystyle \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc) |
![{\displaystyle \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc) |
|
反双曲余弦 |
![{\displaystyle y=\mathrm {arcosh} x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bfa908e6ade964c82ce368b35b859eb6d9077d9b) |
[注 1] |
![{\displaystyle [1,+\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6470ffa3f8825a91419ded64f3654ddbabcb397a) |
![{\displaystyle [0,+\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c0ec7f25cac88c59009a7fe528dc000ec7f58c7) |
|
反双曲正切 |
![{\displaystyle y=\mathrm {artanh} x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/709faafd654b565bc7f2024431c7a776d4a963ce) |
![{\displaystyle {\frac {1}{2}}\ln \left({\frac {1+x}{1-x}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9385c3dc145a78ad4a52ce6006310514f8c9a7e4) |
![{\displaystyle (-1,1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e120a3bd60fc89b495dd7ef6039465b7e6a703b1) |
![{\displaystyle \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc) |
|
反双曲余切 |
![{\displaystyle y=\mathrm {arcoth} x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a2791cbb52daa0e986f51db5a18e73745618496) |
![{\displaystyle {\frac {1}{2}}\ln \left({\frac {x+1}{x-1}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1d5b9d9b258fd9893f62037b9b522b3d4643123) |
![{\displaystyle (-\infty ,-1)\cup (1,+\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d47456502b403eed6fc53e31f4841d330b1df374) |
![{\displaystyle (-\infty ,0)\cup (0,+\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4720a9762dad0001d6c5866ffe76afe33e1bc5a) |
|
反双曲正割 |
![{\displaystyle y=\mathrm {arsech} x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e262d23e293d8cd7da64ab8778e3fd365a3a82a2) |
![{\displaystyle \ln \left({\frac {1}{x}}+{\frac {\sqrt {1-x^{2}}}{x}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9affc36e77408ec923cf90179d2da09b8da38d16) |
![{\displaystyle (0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e70f9c241f9faa8e9fdda2e8b238e288807d7a4) |
![{\displaystyle [0,+\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c0ec7f25cac88c59009a7fe528dc000ec7f58c7) |
|
反双曲余割 |
![{\displaystyle y=\mathrm {arcsch} x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08cb8e227857f2a370864c0d3320f90fecad41fc) |
![{\displaystyle \ln \left({\frac {1}{x}}+{\frac {\sqrt {1+x^{2}}}{\left|x\right|}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83dd95bb9cc86d87cefdca3f6344cff77e084ee2) |
![{\displaystyle (-\infty ,0)\cup (0,+\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4720a9762dad0001d6c5866ffe76afe33e1bc5a) |
![{\displaystyle (-\infty ,0)\cup (0,+\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4720a9762dad0001d6c5866ffe76afe33e1bc5a) |
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反双曲函数的导数
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a8cadb66cdb571835d9d95ff81cb189f43d0ab0)
求导范例:
设θ = arsinh x,则:
![{\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}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7d56ae857b6e2927eee07f6cc12e7fcec483d90)
幂级数展开式
![{\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 }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7306dbcf947ca114605b7c9ac1d90fad12e7a123)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d548a52ebc547d2663ece8b454ae9165c54c1cba)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f2e040edd7ba424a27c7a7b62090afb2e03dc6)
![{\displaystyle =\ln 2x-\sum _{n=1}^{\infty }\left({\frac {(-1)^{n}(2n)!}{2^{2n}(n!)^{2}}}\right){\frac {x^{-2n}}{(2n)}},\qquad x>1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/619a6112dee56bb382e60e82d93c98171ee158d0)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/042385ac6d62082a4ce68602d475b601d45068c4)
![{\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 }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c239e579e423dd7f76809b535a3f0413d20f8587)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a559ff704836a9fdcfdb8812d5f102a70635f59e)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44c227e49955b475b8d15dfe231dda2de950df6c)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/588a3bf00dae48b336c9583df3c09b644506e93f)
![{\displaystyle =x^{-1}+{\frac {x^{-3}}{3}}+{\frac {x^{-5}}{5}}+{\frac {x^{-7}}{7}}+\cdots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7cef86af717cecf1dada61c4f0835a038db53816)
![{\displaystyle =\sum _{n=0}^{\infty }{\frac {x^{-(2n+1)}}{(2n+1)}},\qquad \left|x\right|>1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02a381055841fd8ee3c8f5a00fcd51fbec4a3f13)
![{\displaystyle \operatorname {arcosh} (2x^{2}-1)=2\operatorname {arcosh} x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8301a9ad3904103e4cdc26b3d6bd84913df42b9c)
![{\displaystyle \operatorname {arcosh} (2x^{2}+1)=2\operatorname {arsinh} x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a8a8321c542ccc66ad0657c599824e7de862a34)
反双曲函数的不定积分
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34ee7cca3702151e4eac9357ddaaeb02d212f51f)
使用分部积分法和上面的简单导数很容易得出它们。
注释
- ^ 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。
外部链接
参见