Book

Unlock the elegant connection between conformal mappings and the geometry of the complex plane. In Area Integrals on the Unit Disk, Torang Siregar, S.Pd., Gr., M.Pd., presents a comprehensive and modern treatment of a foundational topic in complex analysis. This work masterfully blends classical theory with modern numerical techniques, employing a "counting-based" approach to illuminate the computation of areas enclosed by curves defined through conformal mappings. Beginning with the historical foundations laid by Smirnov and Gronwall, the book methodically develops the essential tools of the trade: Cauchy's and Green's formulas, Parseval's identity, and the powerful Gronwall Area Theorem. It then guides the reader through a meticulous study of Tchebychev polynomials of the second kind on the ellipse, revealing their remarkable orthogonality with respect to planar measure and their crucial role in approximation theory. From solving fundamental area extremal problems to developing advanced Gaussian quadrature methods, this work bridges the gap between pure mathematics and practical computation. Ideal for graduate students, researchers, and practitioners in complex analysis, approximation theory, and applied mathematics, Area Integrals on the Unit Disk offers a deep, insightful, and practical exploration into a vital area of mathematical analysis.