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:
 * $\operatorname{mod}(\Gamma) := \dfrac 1 {\lambda\left({\Gamma}\right)}$

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 $\operatorname{mod} \left({A}\right)$ 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 $\tilde A = \left\{{z \in \C: r < |z| < R}\right\}$ that is conformally equivalent to $A$.

Then:
 * $\operatorname{mod} \left ({A}\right) := \dfrac 1 {2 \pi} \log \left({\dfrac R r}\right)$

The modulus of $A$ can also be denoted $M \left({R}\right)$.

See Modulus of an Annulus.