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, and , are said to share the value if and have the same solutions (zeros).
More precisely, suppose and are meromorphic functions and are five distinct values. If
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 and as examples, one sees that these two meromorphic functions share the four values but are not equal. So sharing four values is not enough.
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.,
- 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