Bessel Equation in the Semiunbounded Interval x ∈ [x0, ∞]: Solving in the Neighbourhood of an Irregular Singular Point.

This study expresses the solution of the Bessel equation in the neighbourhood of x=∞ as the product of a known-form singular divisor and a specific nonsingular function, which satisfies the corresponding derived equation. Considering the failure of the traditional irregular solution constructed with the power series, we adopt the corrected Fourier series with only limited smooth degree to approximate the nonsingular function in the interval [x0,∞]. In order to guarantee the series’ uniform convergence and uniform approximation to the derived equation, we introduce constraint and compatibility conditions and hence completely determine all undetermined coefficients of the corrected Fourier series. Thus, what we found is not an asymptotic solution at x→∞ (not to mention a so-called formal solution), but a solution in the interval [x0,∞] with certain regularities of distribution. During the solution procedure, there is no limitation on the coefficient property of the equation; that is, the coefficients of the equation can be any complex constant, so that the solution method presented here is universal.