We know that the power series $f(x)= e^x -c \in \mathbb C[[x]]$ for $c \neq 1$, has a multiplicative inverse, since it's constant coefficient is non-zero. I was wondering whether the inverse is known in a closed form. Something like inverse of $\frac{e^x-1}{x}$, which is interesting as it generates the Bernoulli numbers.
2026-05-03 12:01:46.1777809706
Multiplicative inverse of the power series $e^x - c$ for $c \neq 1$.
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Wolfram Alpha can find the inverse in terms of $c$.
The denominators of each term are easy to express in closed form.
The polynomials in the numerators seem to follow Euler's triangle.