Download presentation

Presentation is loading. Please wait.

Published byKatarina Ackroyd Modified over 3 years ago

1
講者： 許永昌 老師 1

2
Contents Prefaces Generating function for integral order Integral representation Orthogonality Reference: http://en.wikipedia.org/wiki/Bessel_function http://en.wikipedia.org/wiki/Bessel_function 2

3
Preface of Bessel function 3 (N) Means only for n Z. R. (N) Means only for n Z. R. (N) Generating function J n (x) Bessel’s ODE series (Ch9.5 ~Ch9.6) Contour integrals (N) Integral representation Recurrence Relations of J. 1 st kind of Bessel function J. 2 nd kind of Bessel function N (or Y ). Orthogonality of J. (N) Integral representati on of N 0. Wronskian

4
Preface of Modified Bessel functions 4 Recurrence Relations of I and K. Modified Bessel functions I and K. Modified Bessel’s ODE Bessel functions J and H . Asymptotic expansion of J , N , I , K. P and Q : for asymptotic

5
Preface of Spherical Bessel functions 5 Recurrence Relations (N) Spherical Bessel functions j n and n n, h n (1) and h n (2). Helmholtz eq. Bessel’s ODE. Bessel functions J, N, H and H . Orthogonality Series forms Limiting values: x << 1 Asymptotic exp. as shown in P4.

6
Generating function for integral order ( 請預讀 P675~P678) (N) Generating function: From this generating function, we can get: J n (x)=(-1) n J n (x)=J n (-x). ------(2) Gotten from g(x,t)=g(-x,t -1 ) Recurrence relations: From x g=(t-1/t)/2*g= J ’ n (x)t n 2J ’ n =J n-1 J n+1 --------(3) J ’ 0 = J 1. From t g=x/2*g*(1+1/t 2 )= J n (x)nt n 1 ---(4) From g(0,t)=g(x,1)=1 J 0 (0)=1,J n (0)=0 & 1=J 0 (x)+2 J 2n (x). By Eqs. (3) & (4) we can get Bessel’s equation. 6

7
Generating function for integral order ( 請預讀 P679~P680) From g(u+v,t)=g(u,t)g(v,t) --(5) (N) Integral representation: From g(x,e i )=exp(ixsin )= J n (x)exp(in ). Therefore, 7

8
Example ( 請預讀 P680~P682) Fraunhofer Diffraction, Circular Aperture: The net light wave will be Based on Eq. (7), we get 8

9
Example (continue) Therefore, when kasin ~3.8317…, | | 0. We get: 9

10
Orthogonality ( 請預讀 P694~P695) m : the m th zero of J . Derivation: Bessel Eq. : 10 x 1 J3(3mx)J3(3mx) Why?

11
Orthogonality (continue) When m n, we get C mn =0. When m=n, use L’Hospital role: 11

12
Orthogonality (continue) Therefore, we get In x [0,1] and f(x=1)=0, m (x)=J ( m x) will form a complete set, because H is a hermitian operator when these boundary condition are held. 12 Q: However, m (0)=0 when 0. Do we need to force the function space obey f(0)=0?

13
Bessel Series (continue) A function may be expanded in 13

14
Homework 11.1.1 (12.1.1e) 11.1.22 (12.1.14e) 11.2.3 14

15
Nouns 15

Similar presentations

OK

Ch 3.4: Complex Roots of Characteristic Equation Recall our discussion of the equation where a, b and c are constants. Assuming an exponential soln leads.

Ch 3.4: Complex Roots of Characteristic Equation Recall our discussion of the equation where a, b and c are constants. Assuming an exponential soln leads.

© 2019 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google