1. Bessel's equation of order n Bessel Equation Solution: Case1: r = n

2. Parametric Bessel Equation Bessel Equation solution

3. Bessel Functions besselj(n,x)

4. 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,'LineWidth',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)')

5. Bessel Functions The first five nonnegative zeros

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

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

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

9. Fourier-Bessel Series The Fourier-Bessel series of a function defined on the interval (0, b) is given by are useful in the evaluation of the coefficients Differential Recurrence Relations Case I: If we choose A2 = 1 and B2 = 0, Case II: If we choose A2 = h > 0, B2 = b Case III: If we choose A2 = 0, B2 = b, n = 0 Square Norm

10. Fourier-Bessel Series