R. I. Badran Bessel's Differential Equation Mathematical Physics
Bessel's Differential Equation:
2 Recalling the DE y n y 0 which has a sinusoidal solution (i.e. sin nx and cos nx) and knowing that these solutions can be treated as power series, we can find a solution to the Bessel's DE which is written as: 2 2 2 x y xy (x p )y 0 . The solution is represented by a series. This series very much look like a damped sine or cosine. It is called a Bessel function.
To solve the Bessel' DE, we apply the Frobenius method by mn assuming a series solution of the form y an x . n0 Substitute this solution into the DE equation and after some mathematical steps you may find that the indicial equation is 2 2 m p 0 where m= p. Also you may get a1=0 and hence all odd a's are zero as well. The recursion relation can be found as
a a n n2 (n m 2)2 p 2 with (n=0, 1, 2, 3, ……etc.).
Case (1): m= p (seeking the first solution of Bessel DE)
We get the solution 2 4 p x x y a x 1 2(2p 2) 2 4(2p 2)(2p 4) This can be rewritten as:
x p2n J (x) y a (1)n p 2n n0 2 n!( p n)! 1 Put a 2 p p! The Bessel function has the factorial form x p2n J (x) (1)n p 2n p , n0 n!( p n)!2 R. I. Badran Bessel's Differential Equation Mathematical Physics
This is because
a a p! a a p! a , a 2 , 2 2(2p 2) 22 ( p 1)! 4 4(2p 4) 2!24 ( p 2)!
a a p! a 4 and so on. 6 6(2p 6) 3!26 ( p 3)! Since (p + n)! = (p + n + 1), and n! = (n + 1), the other form is rewritten as
2n p n 1 x J p (x) (1) n0 (n 1)( p n 1) 2
Case (2): m= -p (seeking the second solution of Bessel DE){sec. 13}
The solution in the factorial form is 2n p n x J p (x) (1) 2n p n0 n!(n p)!2
2n p n 1 x Or J p (x) (1) n0 (n 1)( p n 1) 2 Comments:
i) For integer p: p a. You can show that J-p(x)=(-1) Jp(x). Here J-p (x) is not independent solution. b. It is obvious that we have a problem with J-p (x) when x=0 because this second solution goes to infinity. While the first solution still exist because it is finite. c. When p= 2, the terms in the denominator with n= 0, 1 go to infinity (because The Gamma of a negative integer is infinity), and these terms do not contribute to the sum. Such case does not exist for a positive p.
ii) For nonintegral p: J-p (x) and Jp (x) are two independent solutions and a linear combination of them is a general solution. This linear combination is called Neumann function (termed by Np) or Weber function (termed by Yp). However this function is R. I. Badran Bessel's Differential Equation Mathematical Physics
valid for integral or nonintegral p and is also called the Bessel function of second kind:
cos(p)J p (x) J p (x) N p (x) Yp (x) sin(p)
Important note: It must be noted that this expression is an indeterminate form (0/0) for integral p. However for x 0 it has a limit, which is the correct second solution for integral p.
The best general solution may be written as
y AJ p (x) BN p (x) , where A and B are arbitrary constants. At x = 0 all N's are and the only solution is the Bessel function of first kind Jp (x).
Some properties of Bessel function:
From the factorial form of Jp (x) you may get: x 2 x 4 x6 J (x) 1 for p= 0 (1!)2 22 (2!)2 24 (3!)2 26 x x3 x5 x7 J (x) for p= 1 1 2 (1!)(2!)23 (2!)(3!)25 (3!)(4!)27
dJ (x) From these tow expressions it follows that J (x) . dx 1
Recursion relations for Bessel functions:
d 1. [x p J (x)] x p J (x) dx p p1 d 2. [x p J (x)] x p J (x) dx p p1 2p 3. J (x) J (x) J (x) p1 p1 x p
4. J p1(x) J p1(x) 2Jp (x) p p 5. J (x) J (x) J (x) J (x) J (x) p x p p1 x p p1
R. I. Badran Bessel's Differential Equation Mathematical Physics
Note: Similar relations also hold for Np(x). [Try to prove such relations].
Orthogonality and Normalization of Bessel function:
0 1 if a b xJ p (ax)J p (bx)dx 1 1 1 , 0 J 2 (a) J 2 (a) J 2 (a)) if a b 2 p1 2 p1 2 p where a and b are called zero's of Jp(x).