# Exponential and Logarithm in noncommutative setting

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
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.