Poincaré–Bertrand formula on a piecewise Liapunov curve in two-dimensional Baruch Schneider a, * , Ömer Kavaklıog ˘lu b a Division of Electrophysics Research, Faculty of Sciences and Literature, Department of Mathematics, Izmir University of Economics, Izmir 35330, Turkey b Division of Electrophysics Research, Faculty of Computer Sciences, Department of Computer Engineering, Izmir University of Economics, Izmir 35330, Turkey article info Keywords: Singular integrals Poincaré formula abstract The Poincaré–Bertrand formula for interchanging the order of integration in singular inte- gral operators is generalized to the case of quaternionic singular integrals on a piecewise Liapunov curve in two-dimensional. Ó 2008 Elsevier Inc. All rights reserved. 1. Introduction One of the contemporary research areas of applied mathematics is quaternion analysis. The Cauchy-type integrals and related singular operators have crucial ties to various fields of applied mathematics, mechanics, electrodynamics and general relativity. For this reason, it is consequential to comprehend the basic properties of this integral. ‘Cauchy operators’ should be treated with concessive attention since the order of integration is substantial in these singular integrals. The Poincaré–Bertrand formula for interchanging the order of integration in two repeated Cauchy’s principal integrals states that 1 pi Z Cs ds s  t  1 pi Z Cs 1 f ðs; s 1 Þ s 1  s ds 1 ¼ f ðt; tÞþ 1 pi Z Cs 1 ds 1  1 pi Z Cs f ðs; s 1 Þ ðs  tÞðs 1  sÞ ds; where C is a smooth curve in R 2 , t is a fixed point on C, and f lies on some appropriate function space. The formula has orig- inally published by Hardy [6] and Poincaré [11], and then generalized to accommodate fairly general conditions by various researchers, see in particular [8–10,19]. The formula is of great important in the theory of one-dimensional singular integral equations. In this investigation, we generalize the formula in another direction. The boundary properties of a-hyperholomorphic functions in case of a Liapunov surface have been treated in [7,12,16]. Another class of interesting examples is rectifiable curves. The class of rectifiable curves includes as proper subclasses many other important classes of curves, in particular, smooth (Liapunov) curves, piecewise Liapunov curves and Lipschitz curves. Various properties and applications of the Cauchy-type integral for a-hyperholomorphic functions along rectifiable curves (and domains with rectifiable boundary) can be found, for instance, in [1] and [2]. Let C be a curve in R 2 which contains a finite number of non-smooth points. If the complement (in C) of the union of them is a Liapunov curve, then we will refer to C as a piecewise Liapunov curve in R 2 . In this note, we study the properties of the quaternionic Cauchy-type integrals around a piecewise Liapunov curve. We establish the Sokhotski–Plemelj formulas for the 0096-3003/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2008.03.026 * Corresponding author. E-mail addresses: baruch.schneider@ieu.edu.tr (B. Schneider), omer_kavaklioglu@yahoo.com, omer.kavaklioglu@ieu.edu.tr (Ö. Kavaklıog ˘lu). Applied Mathematics and Computation 202 (2008) 814–819 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc