# Symbolic Computation Applied to Cauchy Type Singular Integrals

^{*}

## Abstract

**:**

## 1. Introduction

`Root`object concept (to represent solutions to one-variable algebraic equations) available in Wolfram Mathematica. This conception of an improved version became possible after the design of the [ASPPlusPMinus] algorithm, created for the rational case, which also calculates the projections associated with the integral of type Cauchy considered in [2], even in cases involving polynomials of the fifth degree or higher. The output provided by these algorithms can be used for several other algorithms to solve singular integral equations, to factorize functions, to compute the dimension of the kernel of some classes of singular integrals and to study the spectra of some operators. Furthermore, since the majority of the concepts and results established for the unit circle within operator theory can be generalized for the real line, we are trying to make the several necessary adaptations in analytical and implementation terms so that the creation of new algorithms that use functions defined in the real line becomes possible. We believe that it will also be possible to extend the methods described in this article to other classes of singular integrals of the Cauchy type, such as those studied in [1,5,6,7,8], at least for the rational case.

## 2. Materials and Methods

#### 2.1. Basic Concepts

#### 2.2. [SInt] Algorithm

- Input $r\left(t\right)$ directly;
- Input the numerator and the poles and multiplicities;
- Input zeros, poles and multiplicities.

#### 2.2.1. [SInt] Algorithm Examples

**Example**

**1.**

**Remark**

**1.**

#### 2.2.2. [SInt] Algorithm: Possible Improvements

- Situation 1: The [SInt] algorithm does not identify whether an inputed function has poles in $\mathbb{T}$.

**Example**

**4.**

**Example**

**5.**

- Situation 2: The [SInt] algorithm does not identify whether valid functions ${x}_{+}$ and ${y}_{-}$ are inputed.

**Example**

**6.**

**Remark**

**3.**

- Situation 3: The [SInt] algorithm is not always efficient with a fifth degree or higher polynomial input.

**Example**

**8.**

**Example**

**9.**

#### 2.3. [ARoots] Algorithm

`Root`object concept (see Figure 28), which allows a precise analysis of the absolute value of the roots (see Figure 29).

#### [ARoots] Algorithm Example

**Example**

**10.**

**Example**

**11.**

`Root`objects (Mathematica uses Root objects to represent solutions of algebraic equations in one variable, when it is impossible to find explicit formulas for these solutions), it is possible to request approximate values. The Root object is not a mere denoting symbol but rather an expression that can be symbolically manipulated and numerically evaluated with any desired precision [1].

#### 2.4. [AZeros] and [APoles] Algorithms

#### [AZeros] and [APoles] Algorithms Examples

**Example**

**12.**

## 3. Results

#### 3.1. [ASPPlusPMinus] Algorithm

`Root`object concept. Thus, some of the incorrect outputs described in Section 2.2, and not considered in the implementation of the [SInt] algorithm [2], do not happen.

#### [ASPPlusPMinus] Algorithm Examples

**Example**

**15.**

**Remark**

**7.**

**Example**

**16.**

**Remark**

**8.**

**Remark**

**9.**

#### 3.2. [SInt]${}_{2.0}$ Algorithm

`Root`object concept and Mathematica’s Solve command. In the case when particular rational functions ${x}_{+}$ and ${y}_{-}$ are inputed, the algorithm also checks if ${x}_{+}$ and $\overline{{y}_{-}}$ have poles at $\mathbb{T}\cup {\mathbb{T}}_{+}$. Thus, the incorrect outputs described in Section 2.2, and not considered in the implementation of the [SInt] algorithm [2], do not happen. However, it cannot validate functions ${x}_{+}$ and ${y}_{-}$, which are not rational functions (Example 22). In this case, the algorithm provides the general expression of the singular integrals ${S}_{\mathbb{T}}X\left(t\right)$ and ${S}_{\mathbb{T}}Y\left(t\right)$ in terms of projection operator ${P}_{+}$ but does not give an incorrect output (as the [SInt] algorithm).

#### [SInt]${}_{2.0}$ Algorithm Examples

**Example**

**18.**

**Remark**

**10.**

**Example**

**19.**

**Remark**

**11.**

**Example**

**20.**

**Remark**

**12.**

**Example**

**21.**

**Remark**

**13.**

`Root`objects.

**Example**

**22.**

**Remark**

**14.**

## 4. Discussion

- We hope that our work within operator theory, and with Mathematica, will help in the path to the future design and implementation of several other analytical algorithms, with numerous applications in many areas of research and technology;
- We are considering the design and implementation of other factorization, spectral and kernel algorithms;
- It is our opinion that the design and implementation of analytical algorithms that work with singular integral operators defined on the real line can constitute a very interesting new line of research;
- We also hope that, going forward, these analytical methods, and their implementation using a computer algebra system with large symbolic and numeric computation capabilities, may contribute to the numerical approach in operator theory.

## Supplementary Materials

_{2.0}.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Conceição, A.C. Symbolic Computation Applied to the Study of the Kernel of Special Classes of Paired Singular Integral Operators. Math. Comput. Sci.
**2021**, 15, 63–90. [Google Scholar] [CrossRef] - Conceição, A.C.; Kravchenko, V.G.; Pereira, J.C. Computing some classes of Cauchy type singular integrals with Mathematica software. Adv. Comput. Math.
**2013**, 39, 273–288. [Google Scholar] [CrossRef] - Conceição, A.C.; Kravchenko, V.G.; Pereira, J.C. Rational functions factorization algorithm: A symbolic computation for the scalar and matrix cases. In Proceedings of the 1st National Conference on Symbolic Computation in Education and Research, Lisboa, Portugal, 2–3 April 2012. [Google Scholar]
- Conceição, A.C.; Kravchenko, V.G.; Pereira, J.C. Factorization Algorithm for Some Special Non-rational Matrix Functions. In Operator Theory: Advances and Applications; Birkhäuser Verlag: Basel, Switzerland, 2010; Volume 202, pp. 87–109. [Google Scholar]
- Conceição, A.C.; Pereira, J.C. Exploring the spectra of some classes of singular integral operators with symbolic computation. Math. Comput. Sci.
**2016**, 10, 291–309. [Google Scholar] [CrossRef] - Conceição, A.C.; Kravchenko, V.G. About explicit factorization of some classes of non-rational matrix functions. Math. Nachr.
**2007**, 280, 1022–1034. [Google Scholar] [CrossRef] - Castro, L.P.; Rojas, E.M.; Saitoh, S.; Tuan, N.M. Solvability of singular integral equations with rotations and degenerate kernels in the vanishing coefficient case. Anal. Appl.
**2015**, 13, 1–21. [Google Scholar] [CrossRef] [Green Version] - Conceição, A.C.; Marreiros, R.C.; Pereira, J.C. Symbolic computation applied to the study of the kernel of a singular integral operator with non-Carleman shift and conjugation. Math. Comput. Sci.
**2016**, 10, 365–386. [Google Scholar] [CrossRef] - Ablowitz, M.J.; Clarkson, P.A. Solitons, Nonlinear Evolution Equations and Inverse Scattering; Cambridge University Press: Cambridge, UK, 1991. [Google Scholar]
- Aktosun, T.; Klaus, M.; van der Mee, C. Explicit Wiener–Hopf factorization for certain non-rational matrix functions. Integral Equ. Oper. Theory
**1992**, 15, 879–900. [Google Scholar] [CrossRef] - Clancey, K.; Gohberg, I. Factorization of Matrix Functions and Singular Integral Operators. In Operator Theory: Advances and Applications; Birkhäuser Verlag: Basel, Switzerland, 1981. [Google Scholar]
- Faddeev, L.D.; Takhatayan, L. Hamiltonian Methods in the Theory of Solitons; Springer: Berlin, Germany, 1987. [Google Scholar]
- Kravchenko, V.G.; Litvinchuk, G.S. Introdution to the Theory of Singular Integral Operators with Shift; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1994. [Google Scholar]
- Litvinchuk, G.S. Solvability Theory of Boundary Value Problems and Singular Integral Equations with Shift; Kluwer Academic Publishers: Dordrecht, The Netherlands, 2000. [Google Scholar]
- Litvinchuk, G.S.; Spitkovskii, I.M. Factorization of Measurable Matrix Functions. In Operator Theory: Advances and Applications; Birkhäuser: Basel, Switzerland, 1987. [Google Scholar]
- Prössdorf, S. Some Classes of Singular Equations; Elsevier: Amsterdam, The Netherlands, 1978. [Google Scholar]
- Gohberg, I.; Krupnik, N. One-Dimensional Linear Singular Integral Equations. In Operator Theory: Advances and Applications; Birkhäuser: Basel, Switzerland, 1992. [Google Scholar]

**Figure 2.**Part of the structure of the [SInt] algorithm corresponding to the Input of ${x}_{+}$ and ${y}_{-}$.

**Figure 5.**Part of the structure of the [SInt] algorithm corresponding to the input of ${x}_{+}$ and ${y}_{-}$.

**Figure 8.**Part of the structure of the [SInt] algorithm corresponding to the input of ${x}_{+}$ and ${y}_{-}$.

**Figure 13.**Part of the structure of the [SInt] algorithm corresponding to the input of ${x}_{+}$ and ${y}_{-}$.

**Figure 16.**Part of the structure of the [SInt] algorithm corresponding to the Input of ${x}_{+}$ and ${y}_{-}$.

**Figure 19.**Part of the structure of the [SInt] algorithm corresponding to the input of ${x}_{+}$ and ${y}_{-}$.

**Figure 22.**Part of the structure of the [SInt] algorithm corresponding to the input of ${x}_{+}$ and ${y}_{-}$.

**Figure 25.**Part of the structure of the [SInt] algorithm corresponding to the input of ${x}_{+}$ and ${y}_{-}$.

**Figure 27.**Part of the code structure of the [ARoot] algorithm responsible for the input options for the polynomial p.

**Figure 28.**Part of the code structure of the [ARoot] algorithm responsible for the computation of the roots of p.

**Figure 29.**Part of the code structure of the [ARoots] algorithm code corresponding to the analysis of the absolute value of the roots of the inputed polynomial.

**Figure 30.**Part of the code structure of the [ARoots] algorithm code corresponding to the computation of an approximate value of a desired root.

**Figure 32.**Part of the structure of the [ARoots] algorithm corresponding to the Option 1 to input p.

**Figure 34.**Part of the structure of the [ARoots] algorithm corresponding to the option to get an approximate root value.

**Figure 38.**Part of the code structure of the [AZeros] algorithm code responsible for the location, relative to the unit circle, of complex numbers in a given list.

**Figure 48.**Part of the code structure of the [ASPPlusPMinus] algorithm that integrates the [APoles] algorithm and uses Mathematica’s Solve command.

**Figure 49.**Part of the structure of the [ASPPlusPMinus] algorithm corresponding to Option 2 to input r.

**Figure 51.**Part of the structure of the [ASPPlusPMinus] algorithm corresponding to Option 1 to input r.

**Figure 53.**Part of the structure of the [ASPPlusPMinus] algorithm corresponding to Option 1 to input r.

**Figure 56.**Part of the code structure of the [SInt]${}_{2.0}$ algorithm responsible for validating ${x}_{+}$.

**Figure 57.**Part of the code structure of the [SInt]${}_{2.0}$ algorithm responsible for the input options.

**Figure 58.**Part of the structure of the [SInt]${}_{2.0}$ algorithm responsible for the input options.

**Figure 60.**Part of the structure of the [SInt]${}_{2.0}$ algorithm responsible for the input options.

**Figure 62.**Part of the structure of the [SInt]${}_{2.0}$ algorithm responsible for the input options.

**Figure 64.**Part of the structure of the [SInt]${}_{2.0}$ algorithm responsible for the input options.

**Figure 66.**Part of the structure of the [SInt]${}_{2.0}$ algorithm responsible for the input options.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Conceição, A.C.; Pires, J.C.
Symbolic Computation Applied to Cauchy Type Singular Integrals. *Math. Comput. Appl.* **2022**, *27*, 3.
https://doi.org/10.3390/mca27010003

**AMA Style**

Conceição AC, Pires JC.
Symbolic Computation Applied to Cauchy Type Singular Integrals. *Mathematical and Computational Applications*. 2022; 27(1):3.
https://doi.org/10.3390/mca27010003

**Chicago/Turabian Style**

Conceição, Ana C., and Jéssica C. Pires.
2022. "Symbolic Computation Applied to Cauchy Type Singular Integrals" *Mathematical and Computational Applications* 27, no. 1: 3.
https://doi.org/10.3390/mca27010003