双曲函数也可以以泰勒级数展开:
![{\displaystyle \sinh 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)!}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/5136eef875e847483a99cd2fdeb3fe99ed38ce76)
![{\displaystyle \cosh x=1+{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}+{\frac {x^{6}}{6!}}+\cdots =\sum _{n=0}^{\infty }{\frac {x^{2n}}{(2n)!}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/ddc1170e1ca7c7a38152fcfe841b60deb418af4f)
![{\displaystyle \tanh x=x-{\frac {x^{3}}{3}}+{\frac {2x^{5}}{15}}-{\frac {17x^{7}}{315}}+\cdots =\sum _{n=1}^{\infty }{\frac {2^{2n}(2^{2n}-1)B_{2n}x^{2n-1}}{(2n)!}},\left|x\right|<{\frac {\pi }{2}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/c95c1e032cc52b10a5e058066523bcd4564f2143)
(罗朗级数)
![{\displaystyle \operatorname {sech} \,x=1-{\frac {x^{2}}{2}}+{\frac {5x^{4}}{24}}-{\frac {61x^{6}}{720}}+\cdots =\sum _{n=0}^{\infty }{\frac {E_{2n}x^{2n}}{(2n)!}},\left|x\right|<{\frac {\pi }{2}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/8b380d5d7c7c0d493b34a9d5d38d9d6b123812a6)
(罗朗级数)
其中
是第
项伯努利数
是第
项欧拉数
![{\displaystyle \int \sinh cx\,\mathrm {d} x={\frac {1}{c}}\cosh cx+C}](//wikimedia.org/api/rest_v1/media/math/render/svg/eacc82af7b3dc13fd697d9a0fad2e8979fb56bc1)
![{\displaystyle \int \cosh cx\,\mathrm {d} x={\frac {1}{c}}\sinh cx+C}](//wikimedia.org/api/rest_v1/media/math/render/svg/aa7d98f04af496b2ea685ab27263e411f08868c4)
![{\displaystyle \int \tanh cx\,\mathrm {d} x={\frac {1}{c}}\ln(\cosh cx)+C}](//wikimedia.org/api/rest_v1/media/math/render/svg/041f85053b1c60628d35290bcc63c9086d58a1c6)
![{\displaystyle \int \coth cx\,\mathrm {d} x={\frac {1}{c}}\ln \left|\sinh cx\right|+C}](//wikimedia.org/api/rest_v1/media/math/render/svg/d5838242353d57832dfd549f3f811bd08e38daf8)
![{\displaystyle \int \operatorname {sech} cx\,\mathrm {d} x={\frac {1}{c}}\arctan(\sinh cx)+C}](//wikimedia.org/api/rest_v1/media/math/render/svg/889a93c9c43a3d07d9fe08f6215e4c74dfc8755a)
![{\displaystyle \int \operatorname {csch} cx\,\mathrm {d} x={\frac {1}{c}}\ln \left|\tanh {\frac {cx}{2}}\right|+C}](//wikimedia.org/api/rest_v1/media/math/render/svg/1d6ceb906c9c036be13352ef4be34522fb619857)
从双曲正弦和馀弦的定义,可以得出如下恒等式:
![{\displaystyle e^{x}=\cosh x+\sinh x}](//wikimedia.org/api/rest_v1/media/math/render/svg/b6cbeacb0a2a95dc7efaa41963fab7b7d9efefe6)
和
![{\displaystyle e^{-x}=\cosh x-\sinh x}](//wikimedia.org/api/rest_v1/media/math/render/svg/b3bc1cd4530b7a249e364da8f1acdc53fa07770f)
反双曲函数是双曲函数的反函数。它们的定义为:
![{\displaystyle {\begin{aligned}\operatorname {arsinh} (x)&=\ln \left(x+{\sqrt {x^{2}+1}}\right)\\\operatorname {arcosh} (x)&=\ln \left(x+{\sqrt {x^{2}-1}}\right);x\geq 1\\\operatorname {artanh} (x)&={\frac {1}{2}}\ln \left({\frac {1+x}{1-x}}\right);\left|x\right|<1\\\operatorname {arcoth} (x)&={\frac {1}{2}}\ln \left({\frac {x+1}{x-1}}\right);\left|x\right|>1\\\operatorname {arsech} (x)&=\ln \left({\frac {1}{x}}+{\frac {\sqrt {1-x^{2}}}{x}}\right);0<x\leq 1\\\operatorname {arcsch} (x)&=\ln \left({\frac {1}{x}}+{\frac {\sqrt {1+x^{2}}}{\left|x\right|}}\right);x\neq 0\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/4037cf41ed88f824727044066df30adc8d80109a)
Ratcliffe, John, Foundations of Hyperbolic Manifolds, Graduate Texts in Mathematics 149, Springer: 99, 2006 [2014-03-27], ISBN 9780387331973, (原始内容存档于2014-01-12), That the area of a hyperbolic triangle is proportional to its angle defect first appeared in Lambert's monograph Theorie der Parallellinien, which was published posthumously in 1786.