# Exceptional value

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A concept in value-distribution theory. Let be a meromorphic function in the whole -plane and let denote its number of -points (counting multiplicities) in the disc . According to R. Nevanlinna's first fundamental theorem (cf. ), as , where is the characteristic function, which does not depend on , is the counting function (the logarithmic average of ) and is a function expressing the average proximity of the values of to on (cf. Value-distribution theory). For the majority of values the quantities and are asymptotically equivalent, as . A (finite or infinite) number is called an exceptional value if this equivalence as is violated. One distinguishes several kinds of exceptional values.

A number is called an exceptional value of in the sense of Poincaré if the number of -points of in the whole plane is finite (cf. , ), in particular if for any .

A number is called an exceptional value of in the sense of Borel if grows slower, in a certain sense, than , as (cf. , ).

A number is called an exceptional value of in the sense of Nevanlinna (cf. ) if its defect (cf. Defective value) A number is called an exceptional value of in the sense of Valiron if A number for which is also called an exceptional value for . The quantity (the positive deviation of ) characterizes the rate of the asymptotic approximation of to (cf. ).

How to Cite This Entry:
Exceptional value. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Exceptional_value&oldid=18994
This article was adapted from an original article by V.P. Petrenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article