Recent Progress in Nevanlinna’s Theory of Meromorphic Functions

Abstract

We shall consider meromorphic functions f defined in the whole complex plane C. If f is a rational function of degree n then f takes on every value a ∈ C*= C u {∞} exactly n times when account is taken of multiplicities and the behavior of f at ∞. When f is transcendental Picard’s theorem (1879) asserts that f takes on every value a ∈ C* infinitely often, with at most two possible exceptions. The exponential function f(z) = ez omits a = O and a = ∞, thus showing that two exceptional values are possible. Since Euler played an important role in the development of ez, we may regard him as one of the important contributions to our subject.