View source for Defective value

{{MSC|30D35}}

''of a meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030620/d0306201.png" />''

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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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

<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>

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" />).

If

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(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.

The number

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(<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.

See also [[Exceptional value|Exceptional value]]; [[Value-distribution theory|Value-distribution theory]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Nevanilinna,   "Analytic functions" , Springer  (1970)  (Translated from German)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W.K. Hayman,   "Meromorphic functions" , Oxford Univ. Press  (1964)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.A. Gol'dberg,   I.V. Ostrovskii,   "Value distribution of meromorphic functions" , Moscow  (1970)  (In Russian)</TD></TR></table>