$\sinh x =\frac{e^x-e^{-x}}{2}= \sum_{n=0}^{\infty} \frac{x^{2n+1}}{(2n+1)!} = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \cdots$
$\cosh x=\frac{e^x+e^{-x}}{2} = \sum_{n=0}^{\infty} \frac{x^{2n}}{(2n)!} = 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \cdots$
$\tanh x =\frac{e^x-e^{-x}}{e^x+e^{-x}}= \sum_{n=1}^{\infty} \frac{B_{2n} 4^n (4^n - 1)}{(2n)!} x^{2n-1} = x - \frac{x^3}{3} + \frac{2x^5}{15} - \cdots$
$\coth x =\frac{e^x+e^{-x}}{e^x-e^{-x}}= \frac{1}{x} + \sum_{n=1}^{\infty} \frac{B_{2n} 4^n}{(2n)!} x^{2n-1} = \frac{1}{x} + \frac{x}{3} - \frac{x^3}{45} + \cdots$
$\operatorname{sech} x=\frac{1}{e^x+e^{-x}} = \sum_{n=0}^{\infty} \frac{E_{2n}}{(2n)!} x^{2n} = 1 - \frac{x^2}{2} + \frac{5x^4}{24} - \cdots$
$\operatorname{csch} x =\frac{1}{e^x-e^{-x}}= \frac{1}{x} - \sum_{n=1}^{\infty} \frac{(2 - 4^n) B_{2n}}{(2n)!} x^{2n-1} = \frac{1}{x} - \frac{x}{6} + \frac{7x^3}{360} - \cdots$
其中：