双曲線正弦関数（ハイパボリックサイン）
\(\displaystyle \sinh x={e^{x}-e^{-x} \over 2}\)
双曲線余弦関数（ハイパボリックコサイン）
\(\displaystyle\cosh x={e^{x}+e^{-x} \over 2}\)
双曲線正接・余接関数
\({\displaystyle \tanh x={\sinh x \over \cosh x},\;\coth x={1 \over \tanh x}}\)
双曲線正割・余割関数
\({\displaystyle \operatorname {sech} x={1 \over \cosh x},\;\operatorname {cosech} x={1 \over \sinh x}}\)
性質
基本性質
\({\displaystyle \cosh ^{2}x-\sinh ^{2}x=1}\)
加法定理
\({\displaystyle {\begin{aligned}\sinh(\alpha +\beta )&=\sinh \alpha \cosh \beta +\cosh \alpha \sinh \beta \\\sinh(\alpha -\beta )&=\sinh \alpha \cosh \beta -\cosh \alpha \sinh \beta \\\cosh(\alpha +\beta )&=\cosh \alpha \cosh \beta +\sinh \alpha \sinh \beta \\\cosh(\alpha -\beta )&=\cosh \alpha \cosh \beta -\sinh \alpha \sinh \beta \\\tanh(\alpha +\beta )&={\frac {\tanh \alpha +\tanh \beta }{1+\tanh \alpha \tanh \beta }}\\\tanh(\alpha -\beta )&={\frac {\tanh \alpha -\tanh \beta }{1-\tanh \alpha \tanh \beta }}\end{aligned}}}\)
微分公式
\({\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} x}}\sinh x&=\cosh x\\{\frac {\mathrm {d} }{\mathrm {d} x}}\cosh x&=\sinh x\\{\frac {\mathrm {d} }{\mathrm {d} x}}\tanh x&=1-\tanh ^{2}x=\operatorname {sech} ^{2}x={\frac {1}{\cosh ^{2}x}}\\{\frac {\mathrm {d} }{\mathrm {d} x}}\coth x&=1-\coth ^{2}x=-\operatorname {csch} ^{2}x=-{\frac {1}{\sinh ^{2}x}}\\{\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {csch} x&=-\coth x\operatorname {csch} x\\{\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {sech} x&=-\tanh x\operatorname {sech} x\end{aligned}}}\)
\({\displaystyle {{\rm {d}}^{2} \over {\rm {d}}x^{2}}y(x)=y(x)}\)
冪級数展開
\({\displaystyle {\begin{aligned}\sinh x&=\sum _{n=0}^{\infty }{\frac {x^{2n+1}}{(2n+1)!}}=x+{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}+{\frac {x^{7}}{7!}}+\dotsb \\[1ex]\cosh x&=\sum _{n=0}^{\infty }{x^{2n} \over (2n)!}=1+{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}+{\frac {x^{6}}{6!}}+\dotsb \\[1ex]\tanh x&=\sum _{n=1}^{\infty }{\frac {2^{2n}(2^{2n}-1)B_{2n}x^{2n-1}}{(2n)!}}=x-{\frac {x^{3}}{3}}+{\frac {2x^{5}}{15}}-{\frac {17x^{7}}{315}}+\dotsb ,\quad |x|<{\frac {\pi }{2}}\\[1ex]\operatorname {csch} x&={\frac {1}{x}}+\sum _{n=1}^{\infty }{\frac {2(1-2^{2n-1})B_{2n}x^{2n-1}}{(2n)!}}={\frac {1}{x}}-{\frac {x}{6}}+{\frac {7x^{3}}{360}}-{\frac {31x^{5}}{15120}}+\dotsb ,\quad 0<|x|<\pi \\[1ex]\operatorname {sech} x&=\sum _{n=0}^{\infty }{\frac {E_{2n}x^{2n}}{(2n)!}}=1-{\frac {x^{2}}{2}}+{\frac {5x^{4}}{24}}-{\frac {61x^{6}}{720}}+\dotsb ,\quad |x|<{\frac {\pi }{2}}\\[1ex]\coth x&={\frac {1}{x}}+\sum _{n=1}^{\infty }{\frac {2^{2n}B_{2n}x^{2n-1}}{(2n)!}}={\frac {1}{x}}+{\frac {x}{3}}-{\frac {x^{3}}{45}}+{\frac {2x^{5}}{945}}+\dotsb ,\quad 0<|x|<\pi \end{aligned}}}\)