The Cauchy-Kowalewski product for bicomplex holomorphic functions (Q2905239)

The authors introduce some elements of function theory basing on the algebra of bicomplex numbers. After some historical remarks and classification of their theory into the set of several function theories the authors introduce in Section 2 some basics for bicomplex numbers and bicomplex holomorphic functions. Moreover, they show the characterization of bicomplex holomorphic functions by a bicomplex variant of Cauchy-Riemann systems. The local representation of bicomplex holomorphic functions in bicomplex power series is shown. After some remarks on algebraic aspects for bicomplex holomorphic functions in Section 3 the authors study in Section 4 the Cauchy-Kowalewsky extension.NEWLINENEWLINE The main results describes the unique continuation of a real analytic function with bicomplex coefficients to a bicomplex holomorphic function in a bicomplex neighborhood of the real axis and vice versa. Finally, in Section 5, they propose a bicomplex Fourier transformation. An application to Hermite polynomials completes the paper.