  We give a formula for the radial asymptotics to all orders of the special
$q$-hypergeometric series known as Nahm sums at complex roots of unity. This
result is used in~\cite{CGZ} to prove one direction of Nahm's conjecture
relating the modularity of Nahm sums to the vanishing of a certain invariant in
$K$-theory. The power series occurring in our asymptotic formula are identical
to the conjectured asymptotics of the Kashaev invariant of a knot once we
convert Neumann-Zagier data into Nahm data, suggesting a deep connection
between asymptotics of quantum knot invariants and asymptotics of Nahm sums
that will be discussed further in a subsequent publication.
