Every open set in the complex plane homeomorphic to an annulus is biholomorphic to exactly one annulus whose inner radius is 1 and whose out radius is $r>1$. Each value of $r$ gives a different complex manifold. This number r is called the "modulus" of the annulus. You could say that the set of real number $(0,\infty)$ give a moduli space of complex annuli.

If I give you a parameterization of the inner boundary and the outer boundary of a topological annulus, is there some explicit formula for the modulus of the annulus it is conformally equivalent to? I am guessing there is some kind of integral formula. I looked through a few papers, and have not found such a formula, but it must certainly be classical.

My question is very similar to

Conformal maps of doubly connected regions to annuli.

but that question has an accepted answer which does not answer my question. The advice on Meta was just to ask a new question, but explain the link to the old one.