Mathematical Physic ylindrical Functions
Mathematical Physics Cylindrical Functions
O Cylindrical Functions Fundamental Properties ◆ Eigenvalue Proble o Symmetric Cylindrical Problems 4 General Cylindrical Problems ◆ Conclusion
Cylindrical Functions Fundamental Properties Eigenvalue Problem Symmetric Cylindrical Problems General Cylindrical Problems Conclusion
Fundamental Properties Cylindrical Functions of order m Definition Special solution of x"+xy+(x-m)y Classification Bessel Function of order m =0!I(k+m+1) Neumann function Jm(x)cosma-_m( of order m sInn Hankel Function of Hn(x)=Jm(x)±iNm(x order m
Fundamental Properties Cylindrical Functions of order m Definition： Classification： Bessel Function of order m Neumann Function of order m Hankel Function of order m " ' ( ) 0 2 2 2 x y +xy + x −m y = ( ) = + + + − = 0 2 2 ! ( 1) ( 1) ( ) k k m x k m k k m J x mx J x mx J x N x m m m sin ( )cos ( ) ( ) − − = Hm(x) = J m(x) i Nm(x) Special solution of
Fundamental Properties Graphs of Cylindrical functions Bessel functions Neumann Functions Properties of Cylindrical Functions Symmetry For mEN, Zm(-x)=(-1)( Asymptotic Properties Null points Recurrence formulas
Fundamental Properties Graphs of Cylindrical Functions – Bessel Functions – Neumann Functions Properties of Cylindrical Functions – Symmetry • For m N, Zm(-x) =(-1)m Zm(x) – Asymptotic Properties – Null points – Recurrence Formulas
Bessel functions .8 0.2 6 0.2 ,4
Bessel Functions
Neumann Functions 0.5 10
Neumann Functions
Asymptotic Properties Asx→0. we have: J0(x)→>1 m>0 (x)→n( N0(x)→>2hn号,Nmo(x)→>-m(2) Asx→o, we have: Jn(x)→2cos(x-mz-是z) m(x)→ Smn(x-m丌-丌 丌X Hm1(x)→√2exl[(x 1m兀 H(x)→、c-(x-号mz一m
Asymptotic Properties As x → 0, we have ： m x m m x x m m m N x N x J x J x ( ) ln , ( ) ( ) ( ) 1, ( ) ( ) ( 1)! 2 2 0 2 0 ! 2 1 0 0 − → → − → → As x → ∞, we have： ( ) exp[ ( )] ( ) exp[ ( )]; ( ) sin( ); ( ) cos( ); 4 1 2 2 1 4 1 2 2 1 4 1 2 2 1 4 1 2 2 1 → − − − → − − → − − → − − − + H x i x m H x i x m N x x m J x x m m x m x m x m x
O Null Points of Bessel Functions According to the graph There are infinitely many positive zeros 0<x (m)<x (m) 2 (m)∠xn+1 The first positive zero increases with the order m 0<x{0)<x(1 2 (m+1) The positive zeros appear alternately (m+1) (m+1) 2 From the asymptotic formula, one obtains cos(x-)mT-It)=0=x-imt-iI=(n-l)I x(m)≈(n+1m-4)z
Null Points of Bessel Functions From the asymptotic formula, one obtains According to the graph： There are infinitely many positive zeros. 0 x1 (m) x2 (m) x3 (m) xn (m) xn (m +1 ) ( ) cos( ) 0 ( ) 4 1 2 ( ) 1 2 1 4 1 2 1 4 1 2 1 + − − − = − − = − x n m x m x m n m n The positive zeros appear alternately. 0 x1 (0) x1 (1) x1 (2) x1 (m) x1 (m+1) The first positive zero increases with the order m. 0 x1 (m) x1 (m+1) x2 (m) x2 (m+1) x3 (m)
Recurrence formulas Basic recurrence formulas [x"zn(x)=-x-"Zm+1(x) X (x)=+x∠m-1(x Corollary 1 Zm'-mZ m /x=-Zm+ m+em/x=+ Corollary 2 2Z=Z1-Z 2mZm/x=Zm-+Zm+l
Recurrence Formulas [ ( )]' ( ) [ ( )]' ( ) 1 1 x Z x x Z x x Z x x Z x m m m m m m m m − + + + − − = + = − Basic recurrence formulas Corollary 2 1 1 ' / ' / − + + = + − = − m m m m m m Z mZ x Z Z mZ x Z Corollary 1 1 1 1 1 2 / 2 ' − + − + = + = − m m m m m m mZ x Z Z Z Z Z
Proof of the recurrence formula x (号)m 从=0k!r(k+m+1) ua(x/x"]=∑ 2k+m 从=0k!r(k+m+ 2k(- k+m 2k-1 k!T(k+m+l =k-1 (-1) 2k+m-12k 从=1(k-1)!r(k+m+1) 1+m+1 0lr(+m+1+1) Jm+1(x)/
Proof of the recurrence formula ( ) = + + + − = 0 2 2 ! ( 1) ( 1) ( ) k k m x k m k k m J x ( ) ( )' ! ( 1) ( 1) [ ( )/ ]' 2 0 2 2 1 k k k m k m m x k k m J x x = + + + − = ( ) 2 1 1 2 2 1 ! ( 1) 2 ( 1) − = + + + − = k k k m k x k k m k ( ) 2 1 1 2 1 2 1 1 ( 1)! ( 1) ( 1) − = + − − − + + − − = k k k m k x k k m l = k −1 ( ) l m m l l m l x x l l m / ! ( 1 1) ( 1) 2 1 0 2 1 2 1 + + = + + + + + − = − m J m (x)/ x = − +1