Applications of the recurrence formulas x n+1 k [xm=+x zm-I=Zo=x [xZi ".d x"xm-[xl-mm-I'dx x"Jm-+(n+m-D)"m-dx x"odx=x"x[x/i]dx Sdx 1+(n-1)x0-(n-1) x m-jax x.+
Applications of the recurrence formulas [ ]' [ ]' [ ]' [ ]' 1 1 1 0 1 1 1 1 x Z x Z Z x xZ x Z x Z Z x x Z m m m m k k k m k m m m − − + + − − − + − − = + = = − = − − − − x J dx = − x x x J m dx n m m m n [ ]' 1 1 1 − − = −x J − + n + m− x J m dx n m n 1 1 1 ( 1) − x J dx = x x xJ dx n n [ ]' 1 1 0 − = x J − n − x J dx n n 1 1 1 ( 1) − − = x J + n − x J − n − x J dx n n n 0 2 2 0 1 1 ( 1) ( 1) x J dx x J m c m m m = + −1
Applications of the recurrence formulas x m-I+(n+m J dx=xm J…+C EX I odx=x Ex 2 3J1+2 0 4 odx Ex 3 Ex. 4 Jo +2 odx 0 Ex 5 xdx=-x,+2j,dx
x J dx = x J + x J − xJ dx 0 0 2 1 3 0 3 2 4 − − x J dx = −x J − + n + m− x J m dx n m n m n 1 1 1 ( 1) 0 1 xJ dx = xJ − − x J dx = x J + n − x J − n − x J dx n n n n 0 2 2 0 1 0 1 ( 1) ( 1) Ex.1 Ex. 2 1 0 J dx = −J Ex. 4 Ex. 3 x J dx = −x J + xJ dx 0 0 2 1 2 2 Ex. 5 xJ2 dx = −xJ1 + 2 J1 dx x J dx x J m c m m m = + −1 Applications of the recurrence formulas
Eigenvalue problems of Bessels Eq The resolution of rotational symmetric cylindrical problems e+ General eigenvalue problems Eigenvalue problems Eigenvalues and eigenfunctions Orthogonality and completeness Typical eigenvalue problems Finiteness and boundary condition of the 1 st kind Finiteness and boundary condition of the 2nd kind Boundary conditions of the I st kind
Eigenvalue problems of Bessel’s Eq. The resolution of rotational symmetric cylindrical problems General eigenvalue problems – Eigenvalue problems – Eigenvalues and eigenfunctions – Orthogonality and completeness Typical eigenvalue problems – Finiteness and boundary condition of the 1st kind – Finiteness and boundary condition of the 2nd kind – Boundary conditions of the 1st kind
The resolution of rotational symmetric cylindrical problem =7(t)R(P) ult-o=f(p)e ampp a△,u T"+a2k2T=0 (pR)-mR+kpr=o T=A,exp(k,aD R=C,Jm,(, )+D, Nn,(k,p) f()=∑4=x,(O(ym
ut a 2 u 2 = ' 0 2 2 T +a k T = im u = T(t)R( )e im t u | f ( )e =0 = exp( ) 2 2 T A k a t = n − n ( ) ( ) n m n n m n R = C J k + D N k = = 1 ( ) ( ) n im n n u T t R e = = 1 ( ) ( ) n An Rn f ( ')' 0 2 2 R − R + k R = m The resolution of rotational symmetric cylindrical problem