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Five-Value Theorem of Nevanlinna
Original post is here eklausmeier.goip.de/blog/2017/06-05-five-value-theorem-of-nevanlinna.
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,
More precisely, suppose
where
then
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
There is also a four-value theorem of Nevanlinna. If two meromorphic functions,
According Frank and Hua (dead link): 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.,
- Gerhard Jank, Lutz Volkmann: Einführung in die Theorie der ganzen und meromorphen Funktionen mit Anwendungen auf Differentialgleichungen
- Lee A. Rubel, James Colliander: Entire and Meromorphic Functions
- Chung-Chun Yang, Hong-Xun Yi: Uniqueness Theory of Meromorphic Functions, five-value theorem proved in §3
For an introduction to complex analysis, see for example Terry Tao:
- 246A, Notes 0: the complex numbers
- 246A, Notes 1: complex differentiation
- 246A, Notes 2: complex integration
- Math 246A, Notes 3: Cauchy’s theorem and its consequences
- Math 246A, Notes 4: singularities of holomorphic functions
- 246A, Notes 5: conformal mapping, covers Picard's great theorem
- 254A, Supplement 2: A little bit of complex and Fourier analysis, proves Poisson-Jensen formula for the logarithm of a meromorphic function in relation to its zeros within a disk