Mathematical Physics Vol 1

5.4 Bessel equation. Bessel functions

235

From the previous relation we obtain

∞ ∑ k = 2 h

k + 2 i =

2 + a

3 +

k + ka

k + a

a 1 x + a 0 x

1 x

k ( k − 1 ) a k x

k x

k x

∞ ∑ k = 2 h ∞ ∑ k = 2 h ∞ ∑ k = 2 {

k i + a

∞ ∑ k = 2

k + ka

2 + a

3 +

k + 2 =

= a 1 x +

k ( k − 1 ) a k x

k x

0 x

1 x

a k x

x k i =

k + ka

k + a

= a 1 x +

k ( k − 1 ) a k x

k x

(5.50)

k − 2

x k = 0 .

= a 1 x +

[ k ( k − 1 )+ k ] a k + a k

− 2 }

In order for this relation to be identically satisfied, it is necessary that the coefficients next to all powers be equal to zero. From this condition it follows that a 1 = 0 , k 2 − k + k a k + a k − 2 = 0 , or a 1 = 0 , a k = − a k − 2 k 2 , za k = 2 , 3 ,... (5.51) From the last relation we can observe that all even coefficients ( k = 2 s + 2 , s = 0 , 1 , 2 ,... ) are expressed in terms of a 0 ,whereas the odd ones ( k = 2 s + 1) are expressed in terms of a 1 . Aswe have obtained that a 1 = 0, it follows that all odd coefficients are equal to zero. For even coefficients we obtain a 0̸ = 0 , a 2 = − a 0 4 , a 4 = − a 2 16 = − 1 16 − a 0 4 = 1 4 · 16 · a 0 , (5.52) etc. By continuing this procedure we obtain the following relations

a 2 s + 1 = 0 ,

a 2 s =

( − 1 ) s ( s ! ) 2 · 2 2 s

(5.53)

a 0 .

If we assume that a 0 = 1, the series r 1 has the form

∞ ∑ s = 0

( − 1 ) s ( s ! ) 2 2 2 s

x 2 s ,

r 1 ( x )=

(5.54)

which converges for each finite x .

Definition The function defined by the relation

( − 1 ) s ( s ! ) 2

x 2

∞ ∑ s = 0

2 s

J 0 ( x )=

(5.55)

is called Bessel function of the first kind of zero order .

Thus, r 1 = J 0 represents the first particular solution of differential equation (5.45).