  Motivated by applications to soft supersymmetry breaking, we revisit the
expansion of the Seiberg-Witten solution around the multi-monopole point on the
Coulomb branch of pure $SU(N)$ $\mathcal{N}=2$ gauge theory in four dimensions.
At this point $N-1$ mutually local magnetic monopoles become massless
simultaneously, and in a suitable duality frame the gauge couplings
logarithmically run to zero. We explicitly calculate the leading threshold
corrections to this logarithmic running from the Seiberg-Witten solution by
adapting a method previously introduced by D'Hoker and Phong. We compare our
computation to existing results in the literature; this includes results
specific to $SU(2)$ and $SU(3)$ gauge theories, the large-$N$ results of
Douglas and Shenker, as well as results obtained by appealing to integrable
systems or topological strings. We find broad agreement, while also clarifying
some lingering inconsistencies. Finally, we explicitly extend the results of
Douglas and Shenker to finite $N$, finding exact agreement with our first
calculation.
