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. ).
|||R. Nevanilinna, "Analytic functions" , Springer (1970) (Translated from German)|
|||A.A. Gol'dberg, I.V. Ostrovskii, "Value distribution of meromorphic functions" , Moscow (1970) (In Russian)|
|||V.P. Petrenko, "Growth of meromorphic functions of finite lower order" Math. USSR Izv. , 3 : 2 (1969) pp. 391–432 Izv. Akad. Nauk SSSR Ser. Mat. , 33 : 2 (1969) pp. 414–454|
An -point of is a point such that .
Exceptional value. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Exceptional_value&oldid=18994