arsinh function

Martin McBride, 2020-02-03
Tags arsinh
Categories hyperbolic functions

The arsinh function is a hyperbolic function. It is the inverse of the sinh, and is also known as the inverse hyperbolic sine function.

Equation and graph

The arsinh function is defined as the inverse of sinh, ie if:

$$ x = \sinh y $$


$$ y = \operatorname{arsinh} x $$

There is a also a formula for finding arsinh directly:

$$ \operatorname{arsinh} x= \ln{x+{\sqrt {x^{2}+1}}} $$

Here is a graph of the function:

arsinh as inverse of sinh

This animation illustrates the relationship between the sinh function and the arsinh function:

The first, blue, curve is the sinh function.

The grey dashed line is the line $y=x$.

The second, red, curve is the arsinh function. As with any inverse function, it is identical to the original function reflected in the line $y=x$.

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