  We consider the Seiberg-Witten solution of pure $\mathcal{N} =2$ gauge theory
in four dimensions, with gauge group $SU(N)$. A simple exact series expansion
for the dependence of the $2 (N-1)$ Seiberg-Witten periods $a_I(u), a_{DI}(u)$
on the $N-1$ Coulomb-branch moduli $u_n$ is obtained around the
$\mathbb{Z}_{2N}$-symmetric point of the Coulomb branch, where all $u_n$
vanish. This generalizes earlier results for $N=2$ in terms of hypergeometric
functions, and for $N=3$ in terms of Appell functions. Using these and other
analytical results, combined with numerical computations, we explore the global
structure of the K\"ahler potential $K = \frac{1}{2\pi} \sum_I \text{Im}(\bar
a_I a_{DI})$, which is single valued on the Coulomb branch. Evidence is
presented that $K$ is a convex function, with a unique minimum at the
$\mathbb{Z}_{2N}$-symmetric point. Finally, we explore candidate walls of
marginal stability in the vicinity of this point, and their relation to the
surface of vanishing K\"ahler potential.
