Computing some classes of Cauchy type singular integrals with Mathematica software
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Computing some classes of Cauchy type singular integrals with Mathematica software Ana C. Conceição · Viktor G. Kravchenko · José C. Pereira
Received: 25 October 2011 / Accepted: 9 August 2012 / Published online: 4 September 2012 © Springer Science+Business Media, LLC 2012
Abstract We constructed an algorithm, [SInt], for computing some classes of Cauchy type singular integrals on the unit circle. The design of [SInt] was focused on the possibility of implementing on a computer all the extensive symbolic and numeric calculations present in the algorithm. Furthermore, we show how the factorization algorithm described in Conceição et al. (2010) allowed us to construct and implement the [SIntAFact] algorithm for calculating several interesting singular integrals that cannot be computed by [SInt]. All the above techniques were implemented using the symbolic computation capabilities of the computer algebra system Mathematica. The corresponding source code of [SInt] is made available in this paper. Several examples of nontrivial singular integrals computed with both algorithms are presented.
Communicated by: Zhongying Chen. Electronic supplementary material The online version of this article (doi:10.1007/s10444-012-9279-7) contains supplementary material, which is available to authorized users. This research was partially supported by CEAF at Instituto Superior Técnico (Portugal). A. C. Conceição (B) · V. G. Kravchenko CEAF, CEDMES, Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade do Algarve, Campus de Gambelas, 8005-139 Faro, Portugal e-mail: [email protected] J. C. Pereira CEAF, CEDMES, Departamento de Engenharia Electrónica e Informática, Faculdade de Ciências e Tecnologia, Universidade do Algarve, Campus de Gambelas, 8005-139 Faro, Portugal e-mail: [email protected]
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Keywords Singular integrals · Symbolic computation · Mathematica software system · Factorization theory Mathematics Subject Classifications (2010) Primary 30E20 · 68W30 · 47A68; Secondary 45E05 · 47A10
1 Introduction In recent years, several software applications were made available to the general public with extensive capabilities of symbolic computation. These applications, known as computer algebra systems (CAS), allow to delegate to a computer all, or a significant part, of the symbolic and numeric calculations present in many mathematical algorithms. In our work we use the computer algebra system Mathematica (http://www.wolfram.com) to implement for the first time on a computer analytical algorithms developed by us and others within the Operator Theory. The main goal of this paper is to present the analytical algorithms [SInt] and [SIntAFact] that compute some classes of Cauchy type singular integrals on the unit circle. Both algorithms were implemented using the symbolic computation capabilities of the computer algebra system Mathematica. Singular integrals are classic mathematical objects with a vast array of applications in the main scientific research areas (see, for instance,
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