Picard theorem

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)

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)