Mathematical Physics Vol 1

5.4 Bessel equation. Bessel functions

239

In this case, using the same procedure, we obtain

y 2 = J

x ) ,

(5.76)

− ν (

where J

− ν is a Bessel function, defined by the expression

− ν =

x 2

( − 1 ) s s ! Γ ( s − ν + 1 ) x

2

− ν ∞

2 s

∑ s = 0

J

(5.77)

.

There is one case remaining, namely when ν is a positive integer, that is, ν = n , where n denotes a natural number. In this case, as the first particular solution we obtain (see Theorem 15, p.233) y 1 = J n ( x ) , (5.78) and as the second particular solution

∞ ∑ k = 0

n

k + CJ

y 2 = x

b k x

n ( x ) log x .

(5.79)

As in the previous examples, the coefficients b k can be determined from equation L ( y 2 )= 0, and we thus obtain

n − 1 ∑ j = 1

1 2 2 j j ! ( n − 1 ) ··· ( n − j )

Ck 0 2

n + b

n

x 2 j −

n −

y 2 ( x )= b 0 x −

0 x −

s n x

∞ ∑ i = 1

C 2

n + 2 i + CJ

k 2 i ( s i + s i + n ) x

n ( x ) log x .

(5.80)

−

In the previous relation

1 2

1 m

s m = 1 +

+ ··· +

,

(5.81)

( − 1 ) i 2 2 i + n i ! ( i + n ) !

b 0 2 n − 1 ( n − 1 ) ! .

k 2 i =

, C = −

In the special case, when C = 1, for b 0 we obtain

n − 1 ( n − 1 ) ! ,

b 0 = − 2

(5.82)

and thus y 2 becomes

1 2 1 2 1 2

x 2

j !

x 2

− n n − 1 ∑ j = 0

( n − j − 1 ) !

2 j

y 2 = −

(5.83)

−

1 n

1 n !

x 2

1 2

n + J n ( x ) log x −

1 +

−

+ ··· +

( − 1 ) i i ! ( i + n ) !

1 i

+ 1 +

1 i + n

x 2

x 2

n ∞

1 2

1 2

2 i

∑ i = 1

1 +

−

+ ··· +

+ ··· +

.