Implicitly defined univariate function

Question

So my fellow numerical computational peeps it may be that I am suffering from sleep deprivation but I'm struggling to numerically compute a function $u \rightarrow h(u)$ defined implicitly as follows:

$$ u^{h(u)} + u^{-h(u)} = a u $$

for some constant $u$. To be clear, that's just saying $u$ raised to $h(u)$ etc, and this should hold for all u. We can just consider $u>0$ and $h(u)>0$ actually. It is clear that for large $u$ we can come close by ignoring the second term on the left. So perhaps the question is best phrased as "how to perturb that case to better solve the above". My instinct was to shove the $u^{-h(u)}$ on the other side, solve for $h(u)$ and iterate - but perhaps someone has a better idea.

Answer 1

I can't help you at the moment with a numeric procedure to find h, but an analytical solution, at least for $u>1$, reads: $$ h=\frac{\mathrm{arcosh}\left(\frac{1}{2}\cdot a \cdot u\right)}{\ln(u)} $$ For $u\in(0,1)$ you can try $u=1/x$ with $x \in (1,\infty)$.

Maybe you can work with that. I'm curious about what you try to model with this thing.