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:<math>\cosh x = 1 + \frac {x^2} {2!} + \frac {x^4} {4!} + \frac {x^6} {6!} + \cdots = \sum_{n=0}^\infty \frac{x^{2n}}{(2n)!}</math>
:<math>\tanh x = x - \frac {x^3} {3} + \frac {2x^5} {15} - \frac {17x^7} {315} + \cdots = \sum_{n=0}^\infty \frac{(-1)^{n-1}2^{2n}(2^{2n}-1)B_nx^{2n-1}}{(2n)!}, \left |x \right | < \frac {\pi} {2} </math>
:<math>\coth x = \frac {1} {x} + \frac {x} {3} - \frac {x^3} {45} + \frac {2x^5} {945} + \cdots = \sum_{n=0}^\infty \frac{(-1)^{n-1}2^{2n} B_n x^{2n-1}} {(2n)!}, 0 < \left |x \right | < \pi </math>
:<math>\operatorname {sech} x = 1 - \frac {x^2} {2} + \frac {5x^4} {24} - \frac {61x^6} {720} + \cdots = \frac {1} {x} + \sum_{n=1}^\infty \frac{(-1)^n E_n x^{2n}}{(2n)!} , \left |x \right | < \frac {\pi} {2} </math>
:<math>\operatorname {csch} x = \frac {1} {x} - \frac {x} {6} +\frac {7x^3} {360} -\frac {31x^5} {15120} + \cdots = \frac {1} {x} + \sum_{n=1}^\infty \frac{(-1)^n 2 (2^{2n}-1) B_n x^{2n-1}}{(2n)!} , 0 < \left |x \right | < \pi </math>
where
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