5th June 2017
In German known as Fünf-Punkte-Satz. This theorem is astounding. It says: If two meromorphic functions share five values ignoring multiplicity, then both functions are equal. Two functions, \(f(z)\) and \(g(z)\), are said to share the value \(a\) if \(f(z) - a = 0\) and \(g(z) - a = 0\) have the same solutions (zeros).
More precisely, suppose \(f(z)\) and \(g(z)\) are meromorphic functions and \(a_1, a_2, \ldots, a_5\) are five distinct values. If
where
then \(f(z) \equiv g(z)\).
For a generalization see Some generalizations of Nevanlinna's five-value theorem. Above statement has been reproduced from this paper.
The identity theorem makes assumption on values in the codomain and concludes that the functions are identical. The five-value theorem makes assumptions on values in the domain of the functions in question.
Taking \(e^z\) and \(e^{-z}\) as examples, one sees that these two meromorphic functions share the four values \(a_1=0, a_2=1, a_3=-1, a_4=\infty\) but are not equal. So sharing four values is not enough.
There is also a four-value theorem of Nevanlinna. If two meromorphic functions, \(f(z)\) and \(g(z)\), share four values counting multiplicities, then \(f(z)\) is a Möbius transformation of \(g(z)\).
According Frank and Hua: We simply say “2 CM + 2 IM implies 4 CM”. So far it is still not known whether “1 CM + 3 IM implies 4 CM"; CM meaning counting multiplicities, IM meaning ignoring multiplicities.
For a full proof there are books, which are unfortunately paywall protected, e.g.,
For an introduction to complex analysis, see for example Terry Tao:
Categories: mathematics
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Author: Elmar Klausmeier