Séries de funções e a desigualdade isoperimétrica

Abstract

The present work investigates the isoperimetric inequality, a mathematical result that relates the area of a region bounded by a closed curve with the length of that curve. Inspired by the legend of Princess Dido, the problem is approached with modern mathematical tools such as Fourier series and Differential Geometry. The work presents fundamental concepts of sequences and series of functions, discussing convergence criteria and properties of power series. Next, it explores the theory of Fourier series with examples, integration, and Parseval’s Identity, establishing the basis for the inequality demonstrations. The central chapters address plane curves and essential elements of Differential Geometry necessary to formulate and demonstrate the theorem. Two main approaches are discussed: one based on Fourier analysis and the other of a geometric nature. The final part of the work delves into the formal theory of Fourier series, including issues of uniqueness, pointwise and uniform convergence, as well as revisiting the study of Parseval’s Identity. The study combines rigorous theory and practical applications, presenting results on series convergence and the isoperimetric inequality itself.