Definition:Modulus (Geometric Function Theory)

Definition
In geometric function theory, the term modulus is used to denote certain conformal invariants of configurations or curve families.

More precisely, the modulus of a curve family $$\Gamma$$ is the reciprocal of its extremal length:
 * $$\operatorname{mod}(\Gamma) := \frac{1}{\lambda(\Gamma)}.$$

Modulus of a Quadrilateral
Consider a quadrilateral; that is, a Jordan domain $$Q$$ in the complex plane (or some other Riemann surface), together with two disjoint closed boundary arcs $$\alpha$$ and $$\alpha'$$.

Then the modulus of the quadrilateral $$Q(\alpha,\alpha')$$ is the extremal length of the family of curves in $$Q$$ that connect $$\alpha$$ and $$\alpha'$$.

Equivalently, there exists a rectangle $$R=\{x+iy: |x|<a, |y|<b\}$$ and a conformal isomorphism between $$Q$$ and $$R$$ under which $$\alpha$$ and $$\alpha'$$ correspond to the vertical sides of $$R$$.

Then the modulus of $$Q(\alpha,\alpha')$$ is equal to the ratio $$a/b$$.

See Modulus of a Quadrilateral.

Modulus of an Annulus
Consider an annulus $$A$$; that is, a domain whose boundary consists of two Jordan curves.

Then the modulus $$\operatorname{mod}(A)$$ is the extremal length of the family of curves in $$A$$ that connect the two boundary components of $$A$$.

Equivalently, there is a round annulus $$\tilde{A}=\{z\in\C: r<|z|<R\}$$ that is conformally equivalent to $$A$$.

Then $$\operatorname{mod}\left ({A}\right) = \frac 1 {2\pi} \log \left({\frac R r}\right)$$.

See Modulus of an Annulus.