nLab
hyperbolic cosine

Definitions

The hyperbolic cosine function is the function cosh:\cosh \;\colon\; \mathbb{R} \to \mathbb{R} from the real numbers to themselves which is characterized by the following condition:

  1. cosh\cosh is the unique solution among smooth functions to the differential equation/initial value problem

    cosh=cosh cosh '' = cosh

    (where a prime indicates the derivative) subject to the initial conditions

    cosh(0) =1 cosh(0) =0. \begin{aligned} cosh(0) &= 1 \\ cosh'(0) & = 0 \,. \end{aligned}

References

See also