BESSEL'S EQUATION AND BESSEL FUNCTIONS
Bessel Equation and Bessel Function: Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are the canonical solutions y(x) of Bessel's differential equation for an arbitrary complex number α, the order of the Bessel function. The differential equation x²y" + xy' + (x² - p²)y = 0 .......(1) Where p is a real constant is called BESSEL'S(1784 _ 1846)EQUATION of order p. It is clear that x=0 is a regular singular point of the equation . Hence we assume a solution of the form ∞ y = Σ cₙxⁿ⁺ʳ .........(2) n=0 where c₀≠0 Substitution of the series for y , y' and y" in equation (1) yields ∞ ∞ Σ (n+r)(n+r-1) cₙxⁿ⁺ʳ + Σ (n+r)cₙxⁿ⁺ʳ n=0 n=0 ∞ ∞ + Σ cₙxⁿ⁺ʳ⁺² - p² Σ cₙxⁿ⁺ʳ =0 n=0