  In 1988 Falk and Randell, based on Arnol'd's 1969 paper on braids, proved
that the pure braid groups are residually nilpotent. They also proved that the
quotients in the lower central series are free abelian groups.
  This brief note uses an example to provide evidence for a much stronger
statement: that each braid $b$ can be written as an infinite sum $b
=\sum_0^\infty b_i$, where each $b_i$ is a linear function of the $i$-th
Vassiliev-Kontsevich $Z_i(b)$ invariant of $b$. The example is pure braids on
two strands. This leads to solving $e^\tau=q$ for $\tau$ a Laurent series in
$q$. We set $\tau = \sum_1^\infty (-1)^{n+1} (q^n - q^{-n})/n$ and use Fourier
series and Parseval's theorem to prove $e^\tau=q$.
  For more than two strands the stronger statement seems to rely on an as yet
unstated Plancherel theorem for braid groups, which is likely both to be both
and to have deep consequences
