Nevanlinna Theory - Defect Relation

Defect Relation

This is one of the main corollaries from the Second Fundamental Theorem. The defect of a meromorphic function at the point a is defined by the formula

By the First Fundamental Theorem, 0 ≤ δ(a,f) ≤ 1, if T(r,f) tends to infinity (which is always the case for non-constant functions meromorphic in the plane). The points a for which δ(a,f) > 0 are called deficient values. The Second Fundamental Theorem implies that the set of deficient values of a function meromorphic in the plane is at most countable and the following relation holds:

where the summation is over all deficient values. This can be considered as a generalization of Picard's theorem. Many other Picard-type theorems can be derived from the Second Fundamental Theorem.

As another corollary from the Second Fundamental Theorem, one can obtain that

which generalizes the fact that a rational function of degree d has 2d − 2 < 2d critical points.