Namespaces
Variants
Actions

Equilibrium relation

From Encyclopedia of Mathematics
Revision as of 17:12, 7 February 2011 by 127.0.0.1 (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

A relation expressing the connection between the growth of a function that is meromorphic for , and its value distribution (see Value-distribution theory). Each meromorphic function has the following equilibrium property: The sum of its counting function , which characterizes the density of the distribution of -points of , and the proximity function , which characterizes the average rate of approximation of to the given number , is invariant for different values of . The equilibrium relation becomes more effective when using the spherical metric.

Let

denote the spherical distance between two numbers and , and, for each complex number , let

where

and let denote the multiplicity of -points of for . As the function differs from the Nevanlinna proximity function by a bounded term. Therefore, on a circle , the function , as before, characterizes the average rate of approximation of to . The following result holds. For each value , , for any complex number in the extended complex plane and for an arbitrary function that is meromorphic in , the equality (the equilibrium relation)

holds, where

and denotes the number of -points of in the disc .

After the foundational work of R. Nevanlinna [1], the equilibrium relation was carried over to -dimensional entire curves (see [3]) and to holomorphic mappings (see [4], [5]).

References

[1] R. Nevanilinna, "Analytic functions" , Springer (1970) (Translated from German)
[2] H. Wittich, "Neueste Ergebnisse über eindeutige analytische Funktionen" , Springer (1955)
[3] H. Weyl, "Meromorphic functions and analytic curves" , Princeton Univ. Press (1943)
[4] B.V. Shabat, "Introduction of complex analysis" , 2 , Moscow (1976) (In Russian)
[5] P. Griffiths, J. King, "Nevanlinna theory and holomorphic mappings between algebraic varieties" Acta Math. , 130 (1973) pp. 145–220


Comments

An -point of a function is a point such that .

The equilibrium relation is often referred to as the "Ahlfors–Shimizu version of Nevanlinna's first main theoremAhlfors–Shimizu version of Nevanlinna's first main theorem" .

See also Nevanlinna theorems and Value-distribution theory for the notions of counting function and proximity function.

References

[a1] P.A. Griffiths, "Entire holomorphic mappings in one and several complex variables" , Annals Math. Studies , 85 , Princeton Univ. Press (1976)
How to Cite This Entry:
Equilibrium relation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equilibrium_relation&oldid=40766
This article was adapted from an original article by V.P. Petrenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article