Bessel Equation: First Frobenius Series Solution around x=0

This lecture covers the solution of the Bessel equation using the series substitution method, focusing on deriving solutions around the regular singular point x=0. Topics include the indicial polynomial, recurrence relations for coefficients, and the derivation of the Bessel function of the first kind. The lecture also explores the gamma function, its definition, recurrence relation, and particular values. Additionally, it discusses the analytical prolongation of the gamma function and the Frobenius series solution to the Bessel equation. Practical applications and normalization of the solutions are highlighted, providing a comprehensive understanding of these mathematical concepts.