On the Laurent series for bicomplex holomorphic functions

The Schwarz lemma in bicomplex analysis

“…The extended set of bicomplex numbers was introduced in Luna-Elizarrarás et al 20 The extended set of bicomplex numbers is…”

“…Once we have this partial order, it is possible to define a bicomplex circumference of hyperbolic radius and center $$$ {Z}_0 $$$: The extended set of bicomplex numbers was introduced in Luna‐Elizarrarás et al 20 The extended set of bicomplex numbers is where $$$ \overline{\mathbb{C}\left(\mathbf{i}\right)} $$$ is the well known compactification of the $$$ \mathbb{C}\left(\mathbf{i}\right) $$$‐complex plane.…”

“…It is necessary to develop the corresponding theories in bicomplex analysis framework comparing with the theories in complex analysis. Luna‐Elizarrarás has been generalized some classical complex analysis theories to bicomplex analysis, for example, bicomplex Riemann mapping theorem, 19 Theorem 8.6.2, bicomplex Weierstrass' theorem, 19, Theorem 10.2.3 bicomplex Abel's theorem, 19, Theorem 10.4.1 bicomplex Cauchy integral theorem, 19, Theorem 11.1.1 bicomplex Borel–Pompeiu formula, 19, Theorem 11.2.2 bicomplex Cauchy integral representation, 19, Theorem 11.2.3 bicomplex Laurent theorem, 20, Theorem 3.2 and bicomplex Residue theorem 21, Theorem 6 . For more details, please see earlier studies research 19–22 and so on.…”

Singularities of bicomplex holomorphic functions

“…Definition 4. Let F be a bicomplex holomorphic function; then, Z 0 is its zero of hyperbolic order k 0 e + m 0 e † with k 0 and m 0 non-negative integers if there exists a bicomplex holomorphic function H such that H(Z 0 ) ∉ 𝔖 0 and (see (1))…”

“…This paper is a continuation of the work 1 where the bicomplex Laurent expansions were introduced for BC-holomorphic functions 𝑓 ∶ Ω ⊂ BC → BC defined on a domain of the form…”