Counting methods of area integrals and Tchebychev polynomials of second kind on the ellipse

TL;DR

This paper explores area calculation methods for various regions like ellipses using conformal mappings and series expansions, and discusses Chebyshev polynomials of the second kind on ellipses for interpolation.

Contribution

It introduces a novel approach combining conformal mapping series expansions with Gronwall's area formula for computing areas of complex regions.

Findings

Derived explicit formulas for areas of ellipses, circles, and lemniscates.

Analyzed orthogonal Chebyshev polynomials of the second kind on ellipses.

Connected conformal mapping techniques with polynomial interpolation methods.

Abstract

We use Gronwall's area formula to find the area of some differents regions as circles, ellipses and lemniscates.We use Laurent and Taylor series expansions of conformal mapping from the exterior of the unit disk to either of these regions to compute the area of them.We close this work with the discussion of orthogonal Tchebychev polynomials of second kind on the ellipse and interpolation.