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