# Bessel's equation of order n y ck x

Bessel's equation of order n y ck x k r Bessel Equation n2 xy' ' y '( x 2 k 0 x ) y 0 2 xy ' ck (k r ) x k r y ' ck (k r ) x k r 1 k 0 k 0 2 x y ' ' xy '( x n ) y 0 y ' ' ck (k r )(k r 1) x k 0 c (k r )(k r 1) x

k r ck ( k r ) x k r k k 0 k 0 c (k r )(k r 1) (k r ) n x k 0 k 0 k 0 n 2 ck x k r ck x k r 2 2 ck 2 x k r k r k k 2 k 0 r 2

2 c0 r n x c1 (r 1) n x 1 r r n, r n Case1: r = n x y ' ' ck ( k r )(k r 1) x k r 2 2 x 2 y ' ' xy '( x 2 n 2 ) y 0 Solution: 2 k r 2 2 ck (k r )(k r 1) (k r ) n x k 2

k r ck 2 x k r k 2 0 0 0 c1 2n 1 0 c1 0 (k n) 2 n 2 ck ck 2 , k (k 2n)ck ck 2 , ck k ( k12 n ) ck 2 , k 2 , 3, 4 , k 2 , 3, 4 , k 2 , 3, 4 , ( 1) k x J n ( x) k ! ( 1 n k ) 2 k 0 2 k n

Parametric Bessel Equation Bessel Equation xy ' ' y '( x n2 x d dx ) y 0 Bessel Equation xy' ( x n2 x ) y 0 solution n 0 xy ' ' y '( x) y 0 J 0 ( x) n 1 xy ' ' y '( x 1x ) y 0 J1 ( x) n 2 xy ' ' y '( x 4x ) y 0 J 2 ( x)

xy' ' y '( x n2 x ) y 0 J n (x) Bessel Functions ( 1) k x J n ( x) k ! ( 1 n k ) 2 k 0 2 k n J n (x) 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,'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)') Bessel Functions The first five nonnegative zeros Bessel Functions ( 1) k x J n ( x) k 0 k!(1 n k ) 2 2 k n Differential Recurrence Relations Properties n 0,1,2, J n ( x) ( 1) n J n ( x) J n ( x) ( 1) n J n ( x) 0 n 0 J n (0) 1 n 0 even function if n is an even integer and an odd function if n is an odd integer. d n x J n ( x) x n J n 1 ( x) dx d n x J n ( x) x n J n 1 ( x) dx

Observe that J 0 ' ( x) J1 ( x) xJ n ' ( x) nJ n ( x) xJ n 1 ( x) Parametric Bessel Equation Parametric Bessel Equation Bessel Equation xy ' ' y '( x n2 x d dx n2 x xy' ( x ) y 0 xy' ' y '(x n2 x ) y 0 ) y 0 d dx n2 x ) y 0 Bessel Equation n 0

n 1 n 2 xy ' ' y '( x) y 0 xy' (x J 0 ( x) n 0 J1 ( x) n 1 xy ' ' y '( x 4x ) y 0 J 2 ( x) n 2 xy ' ' y '( x 1x ) y 0 xy' ' y '( x n2 x ) y 0 J n (x) solution Param Bess Eq solution xy ' ' y '(x) y 0 J 0 ( i x )

xy ' ' y '(x 1x ) y 0 J1 ( i x) xy ' ' y '(x 4x ) y 0 J 2 ( i x) xy ' ' y '(x n2 x ) y 0 J n ( i x) i are defined by means of a boundary condition of the form A2 J n (b) B2 J n ' (b) 0 The eigenvalues of the corresponding Sturm-Liouville problem are 2 i i Parametric Bessel Equation Parametric Bessel Equation xy' ' y '(x n2 x ) y 0 d dx

n2 x ) y 0 xy ' (x The orthogonality relation J n ( i x) i are defined by means of a is orthogonal with respect to the weight function p(x) = x on an interval [0, b] boundary condition of the form A2 J n (b) B2 J n ' (b) 0 The eigenvalues of the corresponding Sturm-Liou ville problem are 2 i i 1,2,3, i Fourier-Bessel Series Fourier-Bessel Series The Fourier-Bessel series of a function defined on the interval (0, b) is given by b f ( x) ci J n ( i x) i 0 ci xJ 0

n ( i x) f ( x)dx J n ( i x) 2 Square Norm 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 Differential Recurrence Relations d n x J n ( x) x n J n 1 ( x) dx d n x J n ( x) x n J n 1 ( x) dx are useful in the evaluation of the coefficients A2 J n (b) B2 J n ' (b) 0 Fourier-Bessel Series Fourier-Bessel Series Fourier-Bessel Series

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