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''of a meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030620/d0306201.png" />''
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$#C+1 = 44 : ~/encyclopedia/old_files/data/D030/D.0300620 Defective value
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A complex number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030620/d0306202.png" /> (finite or infinite) whose defect <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030620/d0306203.png" /> (see below) is positive. Let the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030620/d0306204.png" /> be defined in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030620/d0306205.png" /> of the complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030620/d0306206.png" />. The defect (or deficiency) of the value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030620/d0306207.png" /> is
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030620/d0306208.png" /></td> </tr></table>
+
{{MSC|30D35}}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030620/d0306209.png" /> is Nevanlinna's characteristic function representing the growth of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030620/d03062010.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030620/d03062011.png" />, and
+
''of a meromorphic function $  f $''
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030620/d03062012.png" /></td> </tr></table>
+
A complex number  $  a $(
 +
finite or infinite) whose defect  $  \delta ( a , f  ) $(
 +
see below) is positive. Let the function  $  f $
 +
be defined in the disc  $  | z | < R \leq  \infty $
 +
of the complex plane  $  \mathbf C $.  
 +
The defect (or deficiency) of the value  $  a $
 +
is
  
is the counting function; here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030620/d03062013.png" /> is the number of solutions of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030620/d03062014.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030620/d03062015.png" /> (counted with multiplicity). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030620/d03062016.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030620/d03062017.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030620/d03062018.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030620/d03062019.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030620/d03062020.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030620/d03062021.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030620/d03062022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030620/d03062023.png" /> is a defective value; this equality also holds in some other cases (e.g. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030620/d03062024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030620/d03062025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030620/d03062026.png" />).
+
$$
 +
\delta ( a , f  )  =  1 - \overline{\lim\limits}\; _ {r \rightarrow R } \
 +
 
 +
\frac{N ( r , a , f  ) }{T ( r , f  ) }
 +
,
 +
$$
 +
 
 +
where  $  T ( r , f  ) $
 +
is Nevanlinna's characteristic function representing the growth of  $  f $
 +
for  $  r \rightarrow R $,
 +
and
 +
 
 +
$$
 +
N ( r , a , f  )  =  \int\limits _ { 0 } ^ { r }  [ n ( t , a ) - n
 +
( 0 , a ) ]  d  \mathop{\rm ln}  t + n ( 0 , a )  \mathop{\rm ln}  r
 +
$$
 +
 
 +
is the counting function; here, $  n ( t , a ) $
 +
is the number of solutions of the equation $  f ( z) = a $
 +
in $  | z | \leq  t $(
 +
counted with multiplicity). If $  T ( r) \rightarrow \infty $
 +
as $  r \rightarrow R $,  
 +
then 0 \leq  \delta ( a , f  ) \leq  1 $
 +
for all $  a \in \mathbf C \cup \{ \infty \} $.  
 +
If $  f ( z) \neq a $
 +
for any $  z $,  
 +
then $  \delta ( a , f  )= 1 $
 +
and $  a $
 +
is a defective value; this equality also holds in some other cases (e.g. $  f ( z) = ze  ^ {z} $,  
 +
$  R = \infty $
 +
and $  a = 0 $).
  
 
If
 
If
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030620/d03062027.png" /></td> </tr></table>
+
$$
 +
\overline{\lim\limits}\;
 +
\frac{T ( r , f  ) }{ \mathop{\rm ln}  ( 1 / ( R
 +
- r ) ) }
 +
  = \infty
 +
$$
  
(or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030620/d03062028.png" /> is meromorphic throughout the plane), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030620/d03062029.png" /> (the defect, or deficiency, relation), and the number of defective values for such <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030620/d03062030.png" /> is at most countable. Otherwise, the set of defective values may be arbitrary; thus, for any sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030620/d03062031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030620/d03062032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030620/d03062033.png" />, it is possible to find an entire function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030620/d03062034.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030620/d03062035.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030620/d03062036.png" /> and there are no other defective values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030620/d03062037.png" />. Limitations imposed on the growth of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030620/d03062038.png" /> entail limitations on the defective values and their defects. For instance, a meromorphic function of order zero or an entire function of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030620/d03062039.png" /> cannot have more than one defective value.
+
(or $  f \not\equiv \textrm{ const } $
 +
is meromorphic throughout the plane), then $  \sum _ {a} \delta ( a , f  ) \leq  2 $(
 +
the defect, or deficiency, relation), and the number of defective values for such $  f $
 +
is at most countable. Otherwise, the set of defective values may be arbitrary; thus, for any sequences $  \{ a _  \nu  \} \subset  \mathbf C $
 +
and $  \{ \delta _  \nu  \} \subset  \mathbf R  ^ {+} $,  
 +
$  \sum _  \nu  \delta _  \nu  \leq  1 $,  
 +
it is possible to find an entire function $  f $
 +
such that $  \delta ( a _  \nu  , f  ) = \delta _  \nu  $
 +
for all $  \nu $
 +
and there are no other defective values of $  f $.  
 +
Limitations imposed on the growth of $  f $
 +
entail limitations on the defective values and their defects. For instance, a meromorphic function of order zero or an entire function of order < 1 / 2 $
 +
cannot have more than one defective value.
  
 
The number
 
The number
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030620/d03062040.png" /></td> </tr></table>
+
$$
 +
\Delta ( a , f  )  = 1 - \lim\limits _ {\overline{ {r \rightarrow \infty }}\; } \
 +
 
 +
\frac{N ( r , a , f  ) }{T ( r , f  ) }
 +
 
 +
$$
  
(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030620/d03062041.png" /> meromorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030620/d03062042.png" />) is known as the defect in the sense of Valiron. The set of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030620/d03062043.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030620/d03062044.png" /> may have the cardinality of the continuum, but always has logarithmic capacity zero.
+
( $  f $
 +
meromorphic in $  \mathbf C $)  
 +
is known as the defect in the sense of Valiron. The set of numbers $  a $
 +
for which $  \Delta ( a , f  ) > 0 $
 +
may have the cardinality of the continuum, but always has logarithmic capacity zero.
  
 
See also [[Exceptional value|Exceptional value]]; [[Value-distribution theory|Value-distribution theory]].
 
See also [[Exceptional value|Exceptional value]]; [[Value-distribution theory|Value-distribution theory]].

Latest revision as of 17:32, 5 June 2020


2020 Mathematics Subject Classification: Primary: 30D35 [MSN][ZBL]

of a meromorphic function $ f $

A complex number $ a $( finite or infinite) whose defect $ \delta ( a , f ) $( see below) is positive. Let the function $ f $ be defined in the disc $ | z | < R \leq \infty $ of the complex plane $ \mathbf C $. The defect (or deficiency) of the value $ a $ is

$$ \delta ( a , f ) = 1 - \overline{\lim\limits}\; _ {r \rightarrow R } \ \frac{N ( r , a , f ) }{T ( r , f ) } , $$

where $ T ( r , f ) $ is Nevanlinna's characteristic function representing the growth of $ f $ for $ r \rightarrow R $, and

$$ N ( r , a , f ) = \int\limits _ { 0 } ^ { r } [ n ( t , a ) - n ( 0 , a ) ] d \mathop{\rm ln} t + n ( 0 , a ) \mathop{\rm ln} r $$

is the counting function; here, $ n ( t , a ) $ is the number of solutions of the equation $ f ( z) = a $ in $ | z | \leq t $( counted with multiplicity). If $ T ( r) \rightarrow \infty $ as $ r \rightarrow R $, then $ 0 \leq \delta ( a , f ) \leq 1 $ for all $ a \in \mathbf C \cup \{ \infty \} $. If $ f ( z) \neq a $ for any $ z $, then $ \delta ( a , f )= 1 $ and $ a $ is a defective value; this equality also holds in some other cases (e.g. $ f ( z) = ze ^ {z} $, $ R = \infty $ and $ a = 0 $).

If

$$ \overline{\lim\limits}\; \frac{T ( r , f ) }{ \mathop{\rm ln} ( 1 / ( R - r ) ) } = \infty $$

(or $ f \not\equiv \textrm{ const } $ is meromorphic throughout the plane), then $ \sum _ {a} \delta ( a , f ) \leq 2 $( the defect, or deficiency, relation), and the number of defective values for such $ f $ is at most countable. Otherwise, the set of defective values may be arbitrary; thus, for any sequences $ \{ a _ \nu \} \subset \mathbf C $ and $ \{ \delta _ \nu \} \subset \mathbf R ^ {+} $, $ \sum _ \nu \delta _ \nu \leq 1 $, it is possible to find an entire function $ f $ such that $ \delta ( a _ \nu , f ) = \delta _ \nu $ for all $ \nu $ and there are no other defective values of $ f $. Limitations imposed on the growth of $ f $ entail limitations on the defective values and their defects. For instance, a meromorphic function of order zero or an entire function of order $ < 1 / 2 $ cannot have more than one defective value.

The number

$$ \Delta ( a , f ) = 1 - \lim\limits _ {\overline{ {r \rightarrow \infty }}\; } \ \frac{N ( r , a , f ) }{T ( r , f ) } $$

( $ f $ meromorphic in $ \mathbf C $) is known as the defect in the sense of Valiron. The set of numbers $ a $ for which $ \Delta ( a , f ) > 0 $ 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)
How to Cite This Entry:
Defective value. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Defective_value&oldid=18447
This article was adapted from an original article by E.M. Chirka (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article