We expand the series step-by-step, derive the general recurrence relation, and compute the initial coefficients to construct the general solution. Finally, we show that when $\lambda = n(n+1)$, one of the fundamental solutions elegantly reduces to the Legendre polynomials.

In this comprehensive lecture, we construct the complete general solution to Bessel's differential equation. The analysis begins by applying the Frobenius method around the regular singular point at $x=0$, establishing the indicial equation and the corresponding recurrence relations. We systematically explore the solutions for half-integer and non-integer orders ($\nu$) before tackling the highly complex degenerate case of integer order. For this scenario, where standard Frobenius solutions become linearly dependent, we derive the Neumann function (Bessel function of the second kind, $Y_\nu$) using L'Hôpital's rule. This advanced analytical derivation explicitly addresses the evaluation of limits involving the singularities of the Gamma and Digamma functions, incorporates harmonic sums and the Euler-Mascheroni constant, and ultimately establishes the universally valid true general solution.