テイラー展開
sinh  x = ∑ n = 0 ∞ 1 ( 2 n + 1 ) ! x 2 n + 1 for all x {\displaystyle \sinh x=\sum _{n=0}^{\infty }{\frac {1}{(2n+1)!}}x^{2n+1}\quad {\mbox{ for all }}x}
cosh  x = ∑ n = 0 ∞ 1 ( 2 n ) ! x 2 n for all x {\displaystyle \cosh x=\sum _{n=0}^{\infty }{\frac {1}{(2n)!}}x^{2n}\quad {\mbox{ for all }}x}
tanh  x = ∑ n = 1 ∞ B 2 n 4 n ( 4 n − 1 ) ( 2 n ) ! x 2 n − 1 for | x | < π 2 {\displaystyle \tanh x=\sum _{n=1}^{\infty }{\frac {B_{2n}4^{n}(4^{n}-1)}{(2n)!