closed form inverse of $1/(1-\exp(x)) + 1/x$

Question

In an article I am writing I wrote:

'The expression

$$\mu = \frac{1}{1 - \exp(\lambda)} + \frac{1}{\lambda}$$

does not have a pleasant inverse giving us an expression for $\lambda$ in terms of $\mu$. However, [some stuff on the quality of the approximation $\lambda \approx 1/\mu$ on the domain we are interested in]'

My co-author pointed out that 'pleasant' does not sound very scientific and that we should use 'closed-form' instead. This seems fair enough, but when making the correction I started to doubt if it is true and if there really isn't some sort of complicated closed-form inverse involving Lambert-W-functions or similar.

And then I thought: well if anyone can answer that, it is the nice people at MSE! So, I would be really greatful if someone can confirm and/or deny that there is no closed form expression of $\lambda$ in terms of $\mu$ given the above relation.

The best you could potentially do is express this via some known generalization of the Lambert W function... I've seen a few studied generalizations that come close to matching this form, but couldn't guarentee any would work. Definitely no elementary inverse though. — Brevan Ellefsen, Oct 08 '18 at 22:06

Claude Leibovici · Accepted Answer · 2018-10-08T11:01:00.963

No closed form but, assuming that $\lambda$ is small, then $$\mu=\frac{1}{2}-\frac{\lambda }{12}+\frac{\lambda ^3}{720}-\frac{\lambda ^5}{30240}+\frac{\lambda ^7}{1209600}+O\left(\lambda ^9\right)$$ Now, using series reversion $$\lambda=-12 \left(\mu -\frac{1}{2}\right)-\frac{144}{5} \left(\mu -\frac{1}{2}\right)^3-\frac{19008}{175} \left(\mu -\frac{1}{2}\right)^5-\frac{393984}{875} \left(\mu -\frac{1}{2}\right)^7+O\left(\left(\mu -\frac{1}{2}\right)^9\right)$$

If you prefer something simpler, using the $[2,2]$ Padé approximant $$\mu=\frac{\frac{1}{2}-\frac{\lambda }{12}+\frac{\lambda ^2}{120}} {1+\frac{\lambda ^2}{60}}$$ let you with a quadratic

closed form inverse of $1/(1-\exp(x)) + 1/x$

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