Difference between revisions of "Defective value"
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Revision as of 22:16, 23 November 2014
2020 Mathematics Subject Classification: Primary: 30D35 [MSN][ZBL]
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
).
If
![]() |
(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.
References
[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) |
Defective value. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Defective_value&oldid=18447