Originally Posted by

**Rincewind** Sorry. I still don't see how you get that from

$\displaystyle \arcsin x = y \ln x$

even if you do exchange variables.

I can't see how to express an inverse function of

$\displaystyle y = \frac{\arcsin x}{\ln x}$

I thought about the following...

$\displaystyle y \ln x = \arcsin x$

$\displaystyle \ln x^y = \arcsin x$

$\displaystyle x = \sin\left(\ln x^y\right)$

$\displaystyle x = \frac{1}{2i}\left(x^y - x^{-y}\right)$

so basically x is the root of an equation

$\displaystyle x^y - 2ix - x^{-y} = 0$

which for a general $\displaystyle y$ has no explicit solution.

but maybe I'm missing something...