suivant: Bibliographie monter: A Bloch Constant for précédent: -holomorphy and quasiregularity

# Final remarks

The Picard theorem follows from Bloch's theorem in one variable. It is then interesting to ask whether the same is possible in the case of -holomorphic mappings in . However, here we can directly find a Picard theorem without invoking our Bloch theorem.

Theorem 10 (Picard)   Let . If there are two bicomplex numbers such that is invertible and for which the set

is not in the range of , then is constant.

Proof. We have just to apply the so-called little Picard theorem" [15] to and . The fact that is invertible will insure us that will be equal to a bicomplex number with and nonzero. Let , . Suppose takes the value at . There exist such that . Thus,

Contradiction. Hence, omits . Similarly, omits , omits and omits . Since , and are distinct. Similarly .

In the same way, it is possible to find also a Casorati-Weierstrass theorem:

Theorem 11 (Casorati-Weierstrass)   Let with not identically noninvertible. Then, is dense in .

Proof. The hypotheses imply that we can write with and nonconstant. Then we can apply the Casorati-Weierstrass theorem for to and in order to prove that is dense in .

A famous example of Fatou and Bieberbach (see [10]) shows that the usual formulation of the Picard theorem in does not extend to holomorphic mappings in .

In this connection, we have some interesting consequences of Theorem 11 which can be interpreted as an other kind of little Picard theorem for bicomplex numbers:

Corollary 1   There is no nondegenerate -holomorphic mapping

such that contains a ball.

Corollary 2   Fatou-Bieberbach examples cannot be -holomorphic mappings, i.e. they connot satisfy the complexified Cauchy-Riemann equations.

For a beautiful formulation of Picard's theorem which holds in higher dimensions, see [9]. Also, for a version of Picard's theorem for quasiregular mappings see Rickman [12].

suivant: Bibliographie monter: A Bloch Constant for précédent: -holomorphy and quasiregularity
Dominic Rochon
2000-07-26