Formal Power Series and the Miraculous Relation Between E (exp) and L (log)

Explore the powerful world of formal power series in this 26-minute lecture from the Exploring Research Level Maths series. Delve into a combinatorial approach to analysis that harkens back to Euler and Newton, moving away from traditional function-based methods. Discover the grounded variants of exp and log polyseries, E and L, and their miraculous inverse relationship with respect to composition. Learn how the formal Faulhaber Derivative D plays a crucial role in demonstrating this relationship, connecting to concepts from Algebraic Calculus. Gain insights into the limitations of transcendental functions and the advantages of using formal power series in computational mathematics. This advanced mathematical discussion serves as a precursor to the upcoming Algebraic Calculus Two course, offering a fresh perspective on fundamental concepts in analysis.