David writes: << Can someone give me the a few terms of the power series of the inverse of exp(x)-x at 0? Exact coefficients would be preferred if rational.
The function f(x) := exp(x) - x has f'(0) = 0, so finv := f^(-1) does not possess a power series around 0 in the usual sense. There may be some ambiguity in the question. f(0) = 1, so is the desired series for the inverse function where its argument = 1 ??? Otoh, f(z) = 0 has a set of countably many solutions, symmetric about the x-axis, but with none of them *on* the axis. At any of these fixed points of exp(z), the derivative is nonzero and so the inverse function exists locally. Just as exp(z) has multiple branches of its inverse function, it's likely that exp(z) - z does the same. --Dan