Deriving Inverse Hyperbolic Functions

[MattL]

Just a quick question

Can anyone give a method to derive arcsinh(x) from the definition of sinh(x)?

Thanks

 

[Muzza]

$$\sinh{x} = \frac{e^x - e^{-x}}{2}$$.

Assuming the existence of arcsinh, for every x we must have:

sinh(arcsinh(x)) = x.

For simplicity, let arcsinh(x) = z, so that

$$\sinh(z) = x$$

<=>

$$e^z - e^{-z} = 2x$$

<=>

$$(e^z)^2 - 1 = e^z \cdot 2x$$

That's a quadratic equation in e^z, which can be easily solved.

 

[MattL]

thanks

haven't done that since a-level and had forgotten it completely!

 

[dextercioby]

Since it's a quadratic equation,u'll need to specify the domain.Note that the direct function is defined on all $\mathbb{R}$,while I'm sure u can't say the same about its inverse.

Daniel.

 

[MattL]

I think arcsinh is ok on all of $\mathbb{R}$
With arccosh x has to be greater than or equal to one, but I can't remember the conditions for arctanh