This book offers a historical and analytical study of the origins of holomorphic dynamics—a field linking analytic functions with the behavior of iterated orbits. Central to this account is the 1915–1918 Grand Prix des Sciences Mathématiques, whose winning manuscript extended local research to the entire Riemann sphere and highlighted the striking properties of boundaries between basins of convergence. Viewed broadly, this event marked a convergence of research traditions from the eighteenth and nineteenth centuries—infinitesimal calculus, singular points, infinite series, functional equations, root separation, complex analysis, multi-valued functions, and sequences of functions—each a cornerstone for the general theory that now holds a prominent place in modern analysis. Drawing on unpublished archival material, original French sources, and the intellectual context of early twentieth-century mathematics, the book reconstructs how scientific ambitions and evolving techniques shaped foundational developments in the analysis of complex functions and their iterative dynamics. Blending historical narrative with critique and technical insight, it will appeal to scholars and students interested in the history of mathematics, holomorphic dynamics, and the interplay between ideas and their historical milieu.