04/01/2025
By Bobbie Wu

The Kennedy College of Sciences, Department of Mathematics and Statistics, invites you to attend the Master’s thesis defense by Zachary Phillips on “Close Evaluation of the Cauchy Integral Formula and its Extension to Surface Integrals in Three Dimensions.”

Candidate Name: Zachary Phillips
Degree: Master of Science in Mathematics – Applied & Computational Concentration
Defense Date: Wednesday, April 9, 2025
Time: 2 to 3 p.m.
Location: Southwick Hall, Room 350W

Thesis Title: "Close Evaluation of the Cauchy Integral Formula and its Extension to Surface Integrals in Three Dimensions"

Committee Members:
• Bowei Wu, Ph.D., (Advisor) Department of Mathematics and Statistics, UMass Lowell
• Min Hyung Cho, Ph.D., Department of Mathematics and Statistics, UMass Lowell
• Joris Roos, Ph.D., Department of Mathematics and Statistics, UMass Lowell

Abstract:
The Cauchy integral formula is a classical result in complex analysis that computes the value of an analytic function in a complex domain using values along the boundary of the domain. This formula has been applied to solve mathematical physics problems, where numerical quadrature is required to compute the solutions. One key challenge is to accurately evaluate the Cauchy integral at target locations close to the boundary, a situation that often arises in physical systems when two objects move very close to each other. Classical quadrature methods become numerically unstable and highly inaccurate due to the near-singular behavior of the integrand. This problem was solved in a pioneer work by Ioakimidis, Papadakis, and Perdios, where they used the technique of singularity subtraction to rewrite the near-singular integral into a form that only involves a contour integral of a well-behaved analytic integrand, making it simple to evaluate using a regular quadrature. This produces a “barycentric-type quadrature formula” for the Cauchy integral formula. Unfortunately, this approach does not have a straightforward extension to three dimensions due to the use of complex analysis. This thesis explores a potential generalization of the barycentric quadrature formula to three dimensions using the mathematical tool of geometric calculus (also known as Clifford analysis). We study the representation of the generalized Cauchy integral formula in geometric calculus and present an analysis on the numerical evaluation of surface Cauchy integrals in three dimensions.