Bessel equation: Bessel functions of the 1st kind

This lecture covers the Bessel equation in the context of Ordinary Differential Equations. It starts by introducing the homogeneous linear 2nd order ODE and then focuses on deriving the general solution to the Bessel equation using the series substitution method around the regular singular point. The lecture further discusses the Frobenius-type series solutions associated with the upper and lower roots of the indicial equation, leading to the Bessel functions of the first kind. It also explores the properties and asymptotic forms of Bessel functions, along with examples and recurrence relations. Additionally, it delves into the second linearly independent solution for integer orders and the challenges in obtaining explicit relations for all coefficients.