Defective value
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=34918