Editing Exponential integral (section)

== Inverse function of the Exponential Integral ==
We can express the [[Inverse function]] of the exponential integral in [[power series]] form:<ref>{{Cite web |title=Inverse function of the Exponential Integral {{math|Ei{{sup|-1}}(''x'')}} |url=https://math.stackexchange.com/questions/4901881/inverse-function-of-the-exponential-integral-mathrmei-1x |access-date=2024-04-24 |website=Mathematics Stack Exchange |language=}}</ref>

: <math>\forall |x| < \frac{\mu}{\ln(\mu)},\quad \mathrm{Ei}^{-1}(x) = \sum_{n=0}^\infty \frac{x^n}{n!} \frac{P_n(\ln(\mu))}{\mu^n}</math>

where <math>\mu</math> is the [[Ramanujan–Soldner constant]] and <math>(P_n)</math> is [[polynomial]] sequence defined by the following [[recurrence relation]]:

: <math>P_0(x) = x,\ P_{n+1}(x) = x(P_n'(x) - nP_n(x)).</math>

For <math>n > 0</math>, <math>\deg P_n = n</math> and we have the formula :

: <math>P_n(x) = \left.\left(\frac{\mathrm d}{\mathrm dt}\right)^{n-1} \left(\frac{te^x}{\mathrm{Ei}(t+x)-\mathrm{Ei}(x)}\right)^n\right|_{t=0}.</math>