Cauchy Principal Value Contour Integral with Applications

Matilde Legua,Luis Sánchez-Ruiz

doi:10.3390/e19050215

Matilde Legua, Luis Sánchez-Ruiz

Open Access

https://doi.org/10.3390/e19050215

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Abstract

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.

Highlights

Continuous intrinsic entropy treatment requires the assistance of measure theory
A number of analytical and numerical methods have been developed such as the Glauert and Goldstein methods developed to study improper integrals that appear in this setting, [6,7,8,9,10,11], which are relevant in aerodynamics
We have studied two main ways of extending the Cauchy principal value concept to contour integrals and have considered quite a general case, here called isometric paths, in which both of them coincide

Summary

Introduction

Continuous intrinsic entropy treatment requires the assistance of measure theory. This involves adequate integrals intended to evaluate the unexpected uncertainty, this is the case when differential entropy is studied [1,2,3,4,5] in which case improper integrals are commonly used. Aerodynamic entropy study focuses on the overall performance neglecting singularities that enjoy some symmetric entropic phenomena such as turbulence on wing or rotor blades and for this Cauchy principal values are commonly used [12,13,14]. We revisit this extension, formalize its definition, and provide an useful method for its calculus by residue theory and apply it to some improper integrals involving trigonometric functions including the Glauert method used in thermodynamic study of a 2D wing section.

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Discussion

Conclusion