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Picard's theorem on the behaviour of an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p0726801.png" /> of a complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p0726802.png" /> near an [[Essential singular point|essential singular point]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p0726803.png" /> is a result in classical function theory that is the starting point of numerous profound researches. It consists of two parts: a) Picard's little theorem: Any entire function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p0726804.png" /> assumes any finite complex value with the possible exception of one value; and b) Picard's big theorem: Any single-valued analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p0726805.png" /> assumes any finite complex value, with the possible exception of one value, in an arbitrary neighbourhood around an isolated essential singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p0726806.png" />.
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Picard's theorem on the behaviour of an analytic function  $  f( z) $
 +
of a complex variable $  z $
 +
near an [[Essential singular point|essential singular point]] $  a $
 +
is a result in classical function theory that is the starting point of numerous profound researches. It consists of two parts: a) Picard's little theorem: Any entire function $  f( z) \neq \textrm{ const } $
 +
assumes any finite complex value with the possible exception of one value; and b) Picard's big theorem: Any single-valued analytic function $  f( z) $
 +
assumes any finite complex value, with the possible exception of one value, in an arbitrary neighbourhood around an isolated essential singular point $  a $.
  
 
This theorem was first published by E. Picard ,
 
This theorem was first published by E. Picard ,
  
and it substantially supplements the [[Sokhotskii theorem|Sokhotskii theorem]]. Picard's little theorem is a consequence of the big one. It follows directly from Picard's big theorem that any finite complex value, with the possible exception of one value, is assumed in an arbitrary neighbourhood of an essential singular point infinitely often. For a meromorphic function in the finite plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p0726807.png" />, Picard's theorem takes the form: If the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p0726808.png" /> is essentially singular for a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p0726809.png" /> that is meromorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268010.png" />, then in an arbitrary neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268011.png" /> the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268012.png" /> assumes any complex value in the extended complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268013.png" />, with the possible exception of two values, and moreover infinitely often. The examples of the entire function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268014.png" /> and the meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268015.png" /> show that all these assertions are precise. The exceptional values appearing in Picard's theorem are called Picard exceptional values.
+
and it substantially supplements the [[Sokhotskii theorem|Sokhotskii theorem]]. Picard's little theorem is a consequence of the big one. It follows directly from Picard's big theorem that any finite complex value, with the possible exception of one value, is assumed in an arbitrary neighbourhood of an essential singular point infinitely often. For a meromorphic function in the finite plane $  \mathbf C = \{ {z } : {| z | < \infty } \} $,  
 +
Picard's theorem takes the form: If the point $  a = \infty $
 +
is essentially singular for a function $  F( z) $
 +
that is meromorphic in $  \mathbf C $,  
 +
then in an arbitrary neighbourhood of $  a $
 +
the function $  F( z) $
 +
assumes any complex value in the extended complex plane $  \overline{\mathbf C}\; = \{ {z } : {| z | \leq  \infty } \} $,  
 +
with the possible exception of two values, and moreover infinitely often. The examples of the entire function $  e  ^ {z} \neq 0 $
 +
and the meromorphic function $  \mathop{\rm tan}  z \neq i, - i $
 +
show that all these assertions are precise. The exceptional values appearing in Picard's theorem are called Picard exceptional values.
  
Picard's theorem is substantially supplemented by the [[Iversen theorem|Iversen theorem]] and the [[Julia theorem|Julia theorem]], which show, respectively, that the Picard exceptional values are asymptotic values (cf. [[Asymptotic value|Asymptotic value]]) and that there exist Julia rays <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268016.png" /> starting at the essential singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268017.png" /> and such that the non-exceptional values are taken infinitely often even in an arbitrary small sector having its vertex at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268019.png" /> as symmetry axis.
+
Picard's theorem is substantially supplemented by the [[Iversen theorem|Iversen theorem]] and the [[Julia theorem|Julia theorem]], which show, respectively, that the Picard exceptional values are asymptotic values (cf. [[Asymptotic value|Asymptotic value]]) and that there exist Julia rays $  L $
 +
starting at the essential singular point $  a $
 +
and such that the non-exceptional values are taken infinitely often even in an arbitrary small sector having its vertex at $  a $
 +
and $  L $
 +
as symmetry axis.
  
The following two directions are characteristic in modern studies related to Picard's theorem. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268020.png" /> be the set of essential singular points of a meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268021.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268022.png" /> is a [[Meromorphic function|meromorphic function]] in a certain neighbourhood of any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268023.png" />, and suppose that the cluster set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268024.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268025.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268026.png" /> does not reduce to one value. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268028.png" />, be the set of those values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268029.png" /> that are assumed infinitely often in any neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268030.png" />. Then Picard's theorem asserts that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268031.png" /> is an isolated point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268032.png" />, the complement
+
The following two directions are characteristic in modern studies related to Picard's theorem. Let $  E $
 +
be the set of essential singular points of a meromorphic function $  F( z) $,  
 +
i.e. $  F( z) $
 +
is a [[Meromorphic function|meromorphic function]] in a certain neighbourhood of any point $  z _ {0} \notin E $,  
 +
and suppose that the cluster set $  C( z _ {0} ;  F  ) $
 +
of $  F( z) $
 +
at a point $  z _ {0} \in E $
 +
does not reduce to one value. Let $  R( a;  F) $,  
 +
$  a \in E $,  
 +
be the set of those values $  w \in \overline{\mathbf C}\; $
 +
that are assumed infinitely often in any neighbourhood of $  a $.  
 +
Then Picard's theorem asserts that if $  a $
 +
is an isolated point in $  E $,  
 +
the complement
  
<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/p/p072/p072680/p07268033.png" /></td> </tr></table>
+
$$
 +
CR( a; F  )  = \overline{\mathbf C}\; \setminus  R( a; F  )
 +
$$
  
has the Picard property, i.e. it consists of at most two points. V.V. Golubev established in 1916 that if the capacity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268034.png" /> is zero, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268035.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268036.png" /> has capacity zero for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268037.png" />. It has not been completely determined (up till 1983) what minimal conditions must be imposed on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268038.png" /> in order that the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268039.png" /> has the Picard property for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268040.png" />. Examples show that on the one hand the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268041.png" /> is not sufficient, while on the other that there is a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268043.png" />, outside which there do not exist meromorphic transcendental functions omitting four values , , .
+
has the Picard property, i.e. it consists of at most two points. V.V. Golubev established in 1916 that if the capacity of $  E $
 +
is zero, $  \mathop{\rm cap}  E = 0 $,  
 +
then $  CR( a;  F  ) $
 +
has capacity zero for all $  a \in E $.  
 +
It has not been completely determined (up till 1983) what minimal conditions must be imposed on $  E $
 +
in order that the set $  CR( a;  F  ) $
 +
has the Picard property for all $  a \in E $.  
 +
Examples show that on the one hand the condition $  \mathop{\rm cap}  E= 0 $
 +
is not sufficient, while on the other that there is a set $  E $,
 +
$  \mathop{\rm cap}  E > 0 $,  
 +
outside which there do not exist meromorphic transcendental functions omitting four values , , .
  
The second direction is related to generalizations of Picard's theorem to analytic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268044.png" /> of several complex variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268046.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268047.png" />, Picard's theorem can also be formulated as follows: Any holomorphic mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268048.png" /> that omits at least two points is constant. However, in 1922, P. Fatou constructed an example of a non-singular holomorphic mapping (and even of a biholomorphic mapping) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268049.png" /> for which the set of exceptional values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268050.png" /> contains a non-empty open set. This means that Picard's theorem (and even Sokhotskii's theorem) cannot be generalized directly to the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268051.png" />. Generalizations of Picard's theorem are possible if one starts, for example, from another formulation, which is somewhat artificial for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268052.png" />: Any holomorphic mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268053.png" /> into the complex projective plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268054.png" /> that omits at least three hyperplanes (i.e. points for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268055.png" />) is constant. In particular, Green's theorem applies: Any holomorphic mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268056.png" /> that omits at least <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268057.png" /> hyperplanes in general position is constant (cf. , , ).
+
The second direction is related to generalizations of Picard's theorem to analytic functions $  f( z) $
 +
of several complex variables $  z=( z _ {1} \dots z _ {n} ) $,  
 +
$  n \geq  1 $.  
 +
For $  n= 1 $,  
 +
Picard's theorem can also be formulated as follows: Any holomorphic mapping $  f: \mathbf C \rightarrow \mathbf C $
 +
that omits at least two points is constant. However, in 1922, P. Fatou constructed an example of a non-singular holomorphic mapping (and even of a biholomorphic mapping) $  f: \mathbf C  ^ {2} \rightarrow \mathbf C  ^ {2} $
 +
for which the set of exceptional values $  \mathbf C  ^ {2} \setminus  f( \mathbf C  ^ {2} ) $
 +
contains a non-empty open set. This means that Picard's theorem (and even Sokhotskii's theorem) cannot be generalized directly to the case $  n > 1 $.  
 +
Generalizations of Picard's theorem are possible if one starts, for example, from another formulation, which is somewhat artificial for $  n= 1 $:  
 +
Any holomorphic mapping $  F: \mathbf C \rightarrow \mathbf C P $
 +
into the complex projective plane $  \mathbf C P = \overline{\mathbf C}\; $
 +
that omits at least three hyperplanes (i.e. points for $  n= 1 $)  
 +
is constant. In particular, Green's theorem applies: Any holomorphic mapping $  F: \mathbf C  ^ {m} \rightarrow \mathbf C P  ^ {n} $
 +
that omits at least $  2n+ 1 $
 +
hyperplanes in general position is constant (cf. , , ).
  
Picard's theorem on the uniformization of algebraic curves: If an [[Algebraic curve|algebraic curve]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268058.png" /> has genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268059.png" />, then there exists no pair of meromorphic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268061.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268062.png" />. In other words, uniformization of algebraic curves of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268063.png" /> by means of meromorphic functions is impossible. On the other hand, one can always perform uniformization in the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072680/p07268064.png" /> by means of (meromorphic) elliptic functions.
+
Picard's theorem on the uniformization of algebraic curves: If an [[Algebraic curve|algebraic curve]] $  \Phi ( z, w) = 0 $
 +
has genus $  g > 1 $,  
 +
then there exists no pair of meromorphic functions $  z= f( t) $,  
 +
$  w = h( t) $
 +
such that $  \Phi ( f( t), h( t)) \equiv 0 $.  
 +
In other words, uniformization of algebraic curves of genus $  g > 1 $
 +
by means of meromorphic functions is impossible. On the other hand, one can always perform uniformization in the case $  g= 1 $
 +
by means of (meromorphic) elliptic functions.
  
 
The theorem was established by E. Picard [[#References|[7]]].
 
The theorem was established by E. Picard [[#References|[7]]].
Line 21: Line 97:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  E. Picard,  "Sur une propriété des fonctions entières"  ''C.R. Acad. Sci. Paris'' , '''88'''  (1879)  pp. 1024–1027</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  E. Picard,  "Sur les fonctions entières"  ''C.R. Acad. Sci. Paris'' , '''89'''  (1879)  pp. 662–665</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Picard,  "Mémoire sur les fonctions entières"  ''Ann. Ecole Norm. Sup.'' , '''9'''  (1880)  pp. 145–166</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.V. Shabat,  "Introduction of complex analysis" , '''1–2''' , Moscow  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E.F. Collingwood,  A.J. Lohwater,  "The theory of cluster sets" , Cambridge Univ. Press  (1966)  pp. Chapt. 1;6</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A. Lohwater,  "The boundary behaviour of analytic functions"  ''Itogi Nauk. i Tekhn. Mat. Anal.'' , '''10'''  (1973)  pp. 99–259  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  P. Griffiths,  J. King,  "Nevanlinna theory and holomorphic mappings between algebraic varieties"  ''Acta Math.'' , '''130'''  (1973)  pp. 145–220</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  E. Picard,  "Démonstration d'un théorème général sur les fonctions uniformes liées par une équation algébrique"  ''Acta Math.'' , '''11'''  (1887–1888)  pp. 1–12</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  B.V. Shabat,  "Distribution of values of holomorphic mappings" , Amer. Math. Soc.  (1987)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  E. Picard,  "Sur une propriété des fonctions entières"  ''C.R. Acad. Sci. Paris'' , '''88'''  (1879)  pp. 1024–1027</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  E. Picard,  "Sur les fonctions entières"  ''C.R. Acad. Sci. Paris'' , '''89'''  (1879)  pp. 662–665</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Picard,  "Mémoire sur les fonctions entières"  ''Ann. Ecole Norm. Sup.'' , '''9'''  (1880)  pp. 145–166</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.V. Shabat,  "Introduction of complex analysis" , '''1–2''' , Moscow  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E.F. Collingwood,  A.J. Lohwater,  "The theory of cluster sets" , Cambridge Univ. Press  (1966)  pp. Chapt. 1;6</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A. Lohwater,  "The boundary behaviour of analytic functions"  ''Itogi Nauk. i Tekhn. Mat. Anal.'' , '''10'''  (1973)  pp. 99–259  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  P. Griffiths,  J. King,  "Nevanlinna theory and holomorphic mappings between algebraic varieties"  ''Acta Math.'' , '''130'''  (1973)  pp. 145–220</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  E. Picard,  "Démonstration d'un théorème général sur les fonctions uniformes liées par une équation algébrique"  ''Acta Math.'' , '''11'''  (1887–1888)  pp. 1–12</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  B.V. Shabat,  "Distribution of values of holomorphic mappings" , Amer. Math. Soc.  (1987)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.B. Conway,  "Functions of one complex variables" , Springer  (1978)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.A. Griffiths,  "Entire holomorphic mappings in one and several complex variables" , ''Annals Math. Studies'' , '''85''' , Princeton Univ. Press  (1976)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.B. Conway,  "Functions of one complex variables" , Springer  (1978)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.A. Griffiths,  "Entire holomorphic mappings in one and several complex variables" , ''Annals Math. Studies'' , '''85''' , Princeton Univ. Press  (1976)</TD></TR></table>

Latest revision as of 08:06, 6 June 2020


Picard's theorem on the behaviour of an analytic function $ f( z) $ of a complex variable $ z $ near an essential singular point $ a $ is a result in classical function theory that is the starting point of numerous profound researches. It consists of two parts: a) Picard's little theorem: Any entire function $ f( z) \neq \textrm{ const } $ assumes any finite complex value with the possible exception of one value; and b) Picard's big theorem: Any single-valued analytic function $ f( z) $ assumes any finite complex value, with the possible exception of one value, in an arbitrary neighbourhood around an isolated essential singular point $ a $.

This theorem was first published by E. Picard ,

and it substantially supplements the Sokhotskii theorem. Picard's little theorem is a consequence of the big one. It follows directly from Picard's big theorem that any finite complex value, with the possible exception of one value, is assumed in an arbitrary neighbourhood of an essential singular point infinitely often. For a meromorphic function in the finite plane $ \mathbf C = \{ {z } : {| z | < \infty } \} $, Picard's theorem takes the form: If the point $ a = \infty $ is essentially singular for a function $ F( z) $ that is meromorphic in $ \mathbf C $, then in an arbitrary neighbourhood of $ a $ the function $ F( z) $ assumes any complex value in the extended complex plane $ \overline{\mathbf C}\; = \{ {z } : {| z | \leq \infty } \} $, with the possible exception of two values, and moreover infinitely often. The examples of the entire function $ e ^ {z} \neq 0 $ and the meromorphic function $ \mathop{\rm tan} z \neq i, - i $ show that all these assertions are precise. The exceptional values appearing in Picard's theorem are called Picard exceptional values.

Picard's theorem is substantially supplemented by the Iversen theorem and the Julia theorem, which show, respectively, that the Picard exceptional values are asymptotic values (cf. Asymptotic value) and that there exist Julia rays $ L $ starting at the essential singular point $ a $ and such that the non-exceptional values are taken infinitely often even in an arbitrary small sector having its vertex at $ a $ and $ L $ as symmetry axis.

The following two directions are characteristic in modern studies related to Picard's theorem. Let $ E $ be the set of essential singular points of a meromorphic function $ F( z) $, i.e. $ F( z) $ is a meromorphic function in a certain neighbourhood of any point $ z _ {0} \notin E $, and suppose that the cluster set $ C( z _ {0} ; F ) $ of $ F( z) $ at a point $ z _ {0} \in E $ does not reduce to one value. Let $ R( a; F) $, $ a \in E $, be the set of those values $ w \in \overline{\mathbf C}\; $ that are assumed infinitely often in any neighbourhood of $ a $. Then Picard's theorem asserts that if $ a $ is an isolated point in $ E $, the complement

$$ CR( a; F ) = \overline{\mathbf C}\; \setminus R( a; F ) $$

has the Picard property, i.e. it consists of at most two points. V.V. Golubev established in 1916 that if the capacity of $ E $ is zero, $ \mathop{\rm cap} E = 0 $, then $ CR( a; F ) $ has capacity zero for all $ a \in E $. It has not been completely determined (up till 1983) what minimal conditions must be imposed on $ E $ in order that the set $ CR( a; F ) $ has the Picard property for all $ a \in E $. Examples show that on the one hand the condition $ \mathop{\rm cap} E= 0 $ is not sufficient, while on the other that there is a set $ E $, $ \mathop{\rm cap} E > 0 $, outside which there do not exist meromorphic transcendental functions omitting four values , , .

The second direction is related to generalizations of Picard's theorem to analytic functions $ f( z) $ of several complex variables $ z=( z _ {1} \dots z _ {n} ) $, $ n \geq 1 $. For $ n= 1 $, Picard's theorem can also be formulated as follows: Any holomorphic mapping $ f: \mathbf C \rightarrow \mathbf C $ that omits at least two points is constant. However, in 1922, P. Fatou constructed an example of a non-singular holomorphic mapping (and even of a biholomorphic mapping) $ f: \mathbf C ^ {2} \rightarrow \mathbf C ^ {2} $ for which the set of exceptional values $ \mathbf C ^ {2} \setminus f( \mathbf C ^ {2} ) $ contains a non-empty open set. This means that Picard's theorem (and even Sokhotskii's theorem) cannot be generalized directly to the case $ n > 1 $. Generalizations of Picard's theorem are possible if one starts, for example, from another formulation, which is somewhat artificial for $ n= 1 $: Any holomorphic mapping $ F: \mathbf C \rightarrow \mathbf C P $ into the complex projective plane $ \mathbf C P = \overline{\mathbf C}\; $ that omits at least three hyperplanes (i.e. points for $ n= 1 $) is constant. In particular, Green's theorem applies: Any holomorphic mapping $ F: \mathbf C ^ {m} \rightarrow \mathbf C P ^ {n} $ that omits at least $ 2n+ 1 $ hyperplanes in general position is constant (cf. , , ).

Picard's theorem on the uniformization of algebraic curves: If an algebraic curve $ \Phi ( z, w) = 0 $ has genus $ g > 1 $, then there exists no pair of meromorphic functions $ z= f( t) $, $ w = h( t) $ such that $ \Phi ( f( t), h( t)) \equiv 0 $. In other words, uniformization of algebraic curves of genus $ g > 1 $ by means of meromorphic functions is impossible. On the other hand, one can always perform uniformization in the case $ g= 1 $ by means of (meromorphic) elliptic functions.

The theorem was established by E. Picard [7].

References

[1a] E. Picard, "Sur une propriété des fonctions entières" C.R. Acad. Sci. Paris , 88 (1879) pp. 1024–1027
[1b] E. Picard, "Sur les fonctions entières" C.R. Acad. Sci. Paris , 89 (1879) pp. 662–665
[2] E. Picard, "Mémoire sur les fonctions entières" Ann. Ecole Norm. Sup. , 9 (1880) pp. 145–166
[3] B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) (In Russian)
[4] E.F. Collingwood, A.J. Lohwater, "The theory of cluster sets" , Cambridge Univ. Press (1966) pp. Chapt. 1;6
[5] A. Lohwater, "The boundary behaviour of analytic functions" Itogi Nauk. i Tekhn. Mat. Anal. , 10 (1973) pp. 99–259 (In Russian)
[6] P. Griffiths, J. King, "Nevanlinna theory and holomorphic mappings between algebraic varieties" Acta Math. , 130 (1973) pp. 145–220
[7] E. Picard, "Démonstration d'un théorème général sur les fonctions uniformes liées par une équation algébrique" Acta Math. , 11 (1887–1888) pp. 1–12
[8] B.V. Shabat, "Distribution of values of holomorphic mappings" , Amer. Math. Soc. (1987) (Translated from Russian)

Comments

References

[a1] J.B. Conway, "Functions of one complex variables" , Springer (1978)
[a2] P.A. Griffiths, "Entire holomorphic mappings in one and several complex variables" , Annals Math. Studies , 85 , Princeton Univ. Press (1976)
How to Cite This Entry:
Picard theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Picard_theorem&oldid=48178
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article