Defective value

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of a meromorphic function

A complex number (finite or infinite) whose defect (see below) is positive. Let the function be defined in the disc of the complex plane . The defect (or deficiency) of the value is

where is Nevanlinna's characteristic function representing the growth of for , and

is the counting function; here, is the number of solutions of the equation in (counted with multiplicity). If as , then for all . If for any , then and is a defective value; this equality also holds in some other cases (e.g. , and ).


(or is meromorphic throughout the plane), then (the defect, or deficiency, relation), and the number of defective values for such is at most countable. Otherwise, the set of defective values may be arbitrary; thus, for any sequences and , , it is possible to find an entire function such that for all and there are no other defective values of . Limitations imposed on the growth of entail limitations on the defective values and their defects. For instance, a meromorphic function of order zero or an entire function of order cannot have more than one defective value.

The number

( meromorphic in ) is known as the defect in the sense of Valiron. The set of numbers for which may have the cardinality of the continuum, but always has logarithmic capacity zero.

See also Exceptional value; Value-distribution theory.


[1] R. Nevanilinna, "Analytic functions" , Springer (1970) (Translated from German)
[2] W.K. Hayman, "Meromorphic functions" , Oxford Univ. Press (1964)
[3] A.A. Gol'dberg, I.V. Ostrovskii, "Value distribution of meromorphic functions" , Moscow (1970) (In Russian)
How to Cite This Entry:
Defective value. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by E.M. Chirka (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article