Definition:Modulus (Geometric Function Theory)
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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:
- $\mod \Gamma := \dfrac 1 {\map \lambda \Gamma}$
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 $\map Q {\alpha, \alpha'}$ is the extremal length of the family of curves in $Q$ that connect $\alpha$ and $\alpha'$.
Equivalently, there exists a rectangle $R = \set {x + i y: \cmod x < a, \cmod y < b}$ and a conformal isomorphism between $Q$ and $R$ under which $\alpha$ and $\alpha'$ correspond to the vertical sides of $R$.
Then the modulus of $\map Q {\alpha, \alpha'}$ 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 $\mod A$ 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 $\tilde A = \set {z \in \C: r < \cmod z < R}$ that is conformally equivalent to $A$.
Then:
- $\mod A := \dfrac 1 {2 \pi} \map \ln {\dfrac R r}$
The modulus of $A$ can also be denoted $\map M R$.