Bessels equation of order n Bessel Equation Solution

Bessel's equation of order n Bessel Equation Solution: Case 1: r = n

Parametric Bessel Equation solution

Bessel Functions besselj(n, x)

Bessel Functions X = 0: 0. 1: 20; J = zeros(5, 201); for i = 0: 4 J(i+1, : ) = besselj(i, X); end plot(X, J, 'Line. Width', 1. 5) axis([0 20 -. 5 1]) grid on legend('J_0', 'J_1', 'J_2', 'J_3', ' J_4', 'Location', 'Best') title('Bessel Functions of the First Kind for v = 0, 1, 2, 3, 4') xlabel('X') ylabel('J_v(X)')

Bessel Functions The first five nonnegative zeros

Bessel Functions Differential Recurrence Relations Properties Observe that even function if n is an even integer and an odd function if n is an odd integer.

Parametric Bessel Equation solution Param Bess Eq solution are defined by means of a boundary condition of the form The eigenvalues of the corresponding Sturm-Liouville problem are

Parametric Bessel Equation The orthogonality relation are defined by means of a boundary condition of the form The eigenvalues of the corresponding Sturm-Liou ville problem are is orthogonal with respect to the weight function p(x) = x on an interval [0, b]

Fourier-Bessel Series The Fourier-Bessel series of a function defined on the interval (0, b) is given by Square Norm Case I: If we choose A 2 = 1 and B 2 = 0, Case II: If we choose A 2 = h > 0, B 2 = b Case III: If we choose A 2 = 0, B 2 = b, n = 0 Differential Recurrence Relations are useful in the evaluation of the coefficients

Fourier-Bessel Series