nLab hyperbolic sine

Definitions

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

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

   $$sinh'' = sinh$$

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

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

Related concepts

• hyperbolic cosine

• hyerbolic tangent?

• sine, cosine

References

See also

• Wikipedia, Hyperbolic function