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Consider the power series ring in noncommutative associative formal variables $X, Y$ over the field $\mathbb C$. Let $P$ be a power series. Define the exponential $e^P$ and logarithm $\log (1+P)$ using the usual series expansions. The exponential maps from a neighborhood of $0$ to a neighborhood of $1$ and the logarithm goes the other way.

Let $P$ be such a power series with zero constant term. How to show that the logarithm and exponential maps defined above are inverses of each other?

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Your power series are formal, so what does "neighborhood of X" mean? –  Yuval Filmus Nov 25 '10 at 0:14
    
@Yuval: even on formal power series there is a natural filtration which, at least in the one-variable case, corresponds to a topology induced by the X-adic metric. I am not sure what happens in this case. –  Qiaochu Yuan Nov 25 '10 at 0:34

1 Answer 1

Powers of $P$ commute so you need not worry about the non-commutative underlying variables. Expand $e^{\log(1+P)}$ in powers of $P$ and calculate the power of $P^t$ for each $t$; you'll get $1+P$. That's something like Moebius inversion. You can also do it the opposite way.

In fact, you don't even need to calculate it, since we already know the answer by substitution for the corresponding "real" power series, which do converge in some neighborhood, and the answers must match.

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