Two fundamental theorems, proved by R. Nevanlinna (see , ), that are basic for the theory of value distribution of meromorphic functions (see Value-distribution theory). Let be a meromorphic function on a disc
where means that is meromorphic in the entire open complex plane. For every , , the proximity function of to a number is defined by
and the counting function of the number of -points of by
where denotes the number of -points of , counting multiplicities, in the disc , i.e. the number of elements of , and for , for .
The function is called the Nevanlinna characteristic of .
Nevanlinna's first theorem. For any function that is meromorphic on a disc , for any , , and any complex number ,
Here denotes the first non-zero coefficient in the Laurent expansion about zero of the function if , and of itself if . Thus, for a function whose characteristic increases without limit as , the sum , considered for different values of , is equal to the value up to a bounded additive term . In this sense, all values are equivalent for any function that is meromorphic on . For this reason, the theory of value distribution of meromorphic functions concerns itself with questions about the asymptotic behaviour of one term, or , in the invariant sum (1).
Nevanlinna's second theorem shows that, for almost all points , the principal role in the sum (1) is played by . The statement of the theorem is as follows.
For any function that is meromorphic on a disc , every , , and any distinct numbers in the extended complex plane, the relation
and the term has the following properties:
1) If , i.e. if is meromorphic in the entire open complex plane, then
as , for all values of with the possible exception of a set of finite total measure.
2) If , then
as , for all values of with the possible exception of a set for which
The function is non-decreasing with increasing , and therefore the right-hand term in (2) cannot increase as more rapidly than outside some exceptional set .
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Nevanlinna theorems. V.P. Petrenko (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Nevanlinna_theorems&oldid=13895