  We study an axially symmetric solution of a vortex in the Abelian-Higgs model
at critical coupling in detail. Here we propose a new idea for a perturbative
expansion of a solution, where the winding number of a vortex is naturally
extended to be a real number and the solution is expanded with respect to it
around its origin. We test this idea on three typical constants contained in
the solution and confirm that this expansion works well with the help of the
Pad\'e approximation. For instance, we analytically reproduce the value of the
scalar charge of the vortex with an error of $O(10^{-6})$. This expansion is
also powerful even for large winding numbers.
