The **hyperbolic sine function**, first studied by the 18th century mathematician Jacopo Riccati, is an odd function defined as half of the difference between the exponential functions e^{x} and e^{-x}:

The function defines the hyperbolic sine of an angle. It’s similar to how a sine works for a unit circle, except here we’re working with a hyperbola instead of the unit circle.

The domain and range include real (-∞, ∞) and complex values. For a complex argument, the identity is

sinh(*x*) = −*i*sin(*i*x).

All hyperbolic functions can be written as a series, and the hyperbolic sine function is no exception. It can be written as:

The denominators rapidly increase, which means that higher order terms soon become insignificant. This results in the approximation

sinh(x) ≈ x, x → 0.

## Derivative and Integrals

The derivative of the hyperbolic sine is the hyperbolic cosine function, cosh(x).

Taking the derivative and setting it equal to zero usually gives us a function’s critical points. However, e^{x} is always positive, which means the derivative of sinh x is never zero. Therefore, **the hyperbolic sine doesn’t have any critical points. **

Taking the second derivative and setting it equal to zero gives us the inflection point(s). The second derivative is sinh(x), which equals 0 when e^{x} = e^{-x}, or when x = 0. Therefore, **the hyperbolic sine has one inflection point, at x = 0.**

The integral of sinh(x) is coshx + C.

## Graph of the Hyperbolic Sine Function

The graph of the hyperbolic sine function is shown below:

It is a monotonic function, unlike its trigonometric relative the sine function, which is a periodic function.

## Identities

The hyperbolic sine has many useful identities, including:

- sinh(-x) = -sinh (x)
- sinh (x + y) = sinh (x) cosh(y) + cosh(x) sinh(y)
- sinh(x) + cosh(x) = e
^{x} - sinh(x) – cosh(x) = -e
^{x} - cosh
^{2}(x) – sinh^{2}(x) = 1

## References

Graph created with Desmos.com.