On factorization of bicomplex meromorphic functions. (English) Zbl 1168.30024
Sabadini, Irene (ed.) et al., Hypercomplex analysis. Selected papers of the 6th ISAAC conference, Ankara, Turkey, August 13–18, 2007. Basel: Birkhäuser (ISBN 978-3-7643-9892-7/hbk; 978-3-7643-9893-4/e-book). Trends in Mathematics, 55-68 (2009).
Summary: In this paper the factorization theory of meromorphic functions of one complex variable is promoted to bicomplex meromorphic functions. Many results of the one complex variable case are seen to hold in the bicomplex case, and it is found that there are results for meromorphic functions of one complex variable which are not true for bicomplex meromorphic functions. In particular, we show that for any bicomplex transcendental meromorphic function \(F\), there exists a bicomplex meromorphic function \(G\) such that \(GF\) is prime even if the set
\[
\{ a \in \mathbb{T}: F(w) + a\varphi (w) \text{ is not prime}\}
\]
is empty or of cardinality \(\mathfrak N_1\) for any non-constant fractional linear bicomplex function \(\varphi\). Moreover, as a specific application, we obtain six additional possible forms of factorization of the complex cosine cos \(z\) in the bicomplex space.
For the entire collection see [Zbl 1159.30003].
For the entire collection see [Zbl 1159.30003].
MSC:
| 30G35 | Functions of hypercomplex variables and generalized variables |
| 30D30 | Meromorphic functions of one complex variable (general theory) |
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