nLab
hyperbolic sine

Definitions

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

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

    sinh=sinh sinh'' = sinh

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

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

References

See also