Abstract
The aim of this paper is to generalize the Landau‐type Tauberian theorem for the bicomplex variables. Our findings extend and improve on previous versions of the Ikehara theorem. Also boundedness result for the bicomplex version of Ikehara–Korevaar theorem is derived. The purpose of this article is to substantially extend the various complex Tauberian theorems for the Dirichlet series to the bicomplex domain.
Highlights
IntroductionBicomplex numbers have been investigated, and a lot of work has been carried out in this area
For a long time, bicomplex numbers have been investigated, and a lot of work has been carried out in this area
Bicomplex numbers are introduced by Segre [1] in 1882
Summary
Bicomplex numbers have been investigated, and a lot of work has been carried out in this area. E aim of this paper is to extend the various complex Tauberian theorems for the Dirichlet series to the bicomplex domain. Ringleb [24] (see [22]), investigated the analyticity of a bicomplex function with respect to its idempotent complex component functions in the following theorem. Tauberian type theorems have numerous applications in mathematics, including rapidly decaying distributions and their applications to stable laws [30], generalized functions [31], Dirichlet series [32], and the solution of the prime number theorem [26]. Landau [35] (see [[32], p.4]) studied the following Tauberian result for complex power series. Is the region of convergence of the bicomplex Dirichlet series f(ξ) defined in equation (26). Inspired by the work of Agarwal et al [10] and Srivastava and Kumar [37], here, the bicomplex Landau-type Tauberian theorem is investigated. The bicomplex version of the Ikehara’s Tauberian theorem, which is generalization of the Landau-type Tauberian theorem, has been studied
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