Cauchy Principal Value Contour Integral with Applications

[EN] Cauchy principal value is a standard method applied in mathematical applications by which an improper, and possibly divergent, integral is measured in a balanced way around singularities or at infinity. On the other hand, entropy prediction of systems behavior from a thermodynamic perspective commonly involves contour integrals. With the aim of facilitating the calculus of such integrals in this entropic scenario, we revisit the generalization of Cauchy principal value to complex contour integral, formalize its definition and-by using residue theory techniques-provide an useful way to evaluate them.Legua Fernandez, MP.; Sánchez Ruiz, LM. (2017). Cauchy Principal Value Contour Integral with Applications. Entropy. 19(5):1-9. doi:10.3390/e19050215S1919

An asymptotic expansion for product integration applied to Cauchy principal value integrals

Product integration methods for Cauchy principal value integrals based on piecewise Lagrangian interpolation are studied. It is shown that for this class of quadrature methods the truncation error has an asymptotic expansion in integer powers of the step-size, and that a method with an asymptotic expansion in even powers of the step-size does not exist. The relative merits of a quadrature method which employs values of both the integrand and its first derivative and for which the truncation error has an asymptotic expansion in even powers of the step-size are discussed

Numerical Evaluation of a Generalized Cauchy Principal Value

A method is presented for the evaluation of a generalized Cauchy principal value of an improper integral. The integral is transformed into a Riemann integral which is evaluated using Romberg integration. This method is very convenient for computational purposes.The classical Cauchy principal value can be considered as a special case of the present one

Sigmoidal--trapezoidal quadrature for ordinary and Cauchy principal value integrals

Consider the evaluation of If:=^^f201f(x) dx . Among all the quadrature rules for the approximate evaluation of this integral, the trapezoidal rule is known for its simplicity of construction and, in general, its slow rate of convergence to If. However, it is well known, from the Euler-Maclaurin formula, that if f is periodic of period 1, then the trapezoidal rule can converge very quickly to If. A sigmoidal transformation is a mapping of [0,1] onto itself and is such that when applied to If gives an integrand having some degree of periodicity. Applying the trapezoidal rule to the transformed integral gives an increased rate of convergence. In this paper, we explore the use of such transformations for both ordinary and Cauchy principal value integrals. By considering the problem in a suitably weighted Sobolev space, a very satisfactory analysis of the rate of convergence of the truncation error is obtained. This combination of a sigmoidal transformation followed by the trapezoidal rule gives rise to the so-called sigmoidal-trapezoidal quadrature rule of the title

Non-integer valued winding numbers and a generalized Residue Theorem

We define a generalization of the winding number of a piecewise $C^1$ cycle in the complex plane which has a geometric meaning also for points which lie on the cycle. The computation of this winding number relies on the Cauchy principal value, but is also possible in a real version via an integral with bounded integrand. The new winding number allows to establish a generalized residue theorem which covers also the situation where singularities lie on the cycle. This residue theorem can be used to calculate the value of improper integrals for which the standard technique with the classical residue theorem does not apply.Comment: Final version, 19 pages, 7 figure