Mathematical Physics Vol 1

Chapter 5. Series Solutions of Differential Equations. Special functions

242

Figure 5.4: Weber function.

In the special case, for n = 0we obtain

( − 1 ) k

x 2

2 k

∞ ∑ k = 0

Γ ′ ( k + 1 ) Γ ( k + 1 )

2 π

x 2 −

2 π

Y 0 ( x )=

J 0 ( x ) ln

(5.97)

.

( k ! ) 2

The functions J ν and Y ν are linearly independent, and for every ν (integer or not integer) they form the fundamental solution of the initial equation. The general solution can now be represented in the form y = C 1 J ν ( x )+ C 2 Y ν ( x ) , (5.98) where C i ( i = 1 , 2 ) are arbitrary constants. Let us now write some recurrence formulas for the Bessel and Weber functions. J ′ ν ( x )= J ν − 1 ( x ) − ν x J ν ( x ) , (5.99) Y ′ ν ( x )= Y ν − 1 ( x ) − ν x Y ν ( x ) , (5.100) J ′ ν ( x )= − J ν + 1 ( x )+ ν x J ν ( x ) , (5.101) Y ′ ν ( x )= − Y ν + 1 ( x )+ ν x Y ν ( x ) , (5.102) J ν + 1 ( x )= 2 ν x J ν ( x ) − J ν − 1 ( x ) , (5.103) Y ν + 1 ( x )= 2 ν x Y ν ( x ) − Y ν − 1 ( x ) . (5.104) These formulas can be verified by differentiating the Bessel and Weber functions. Let us demonstrate this on the example (5.99)

∞ ∑ k = 0

( − 1 ) k 2 ( ν + k ) x 2 ν + 2 k − 1 2 ν + 2 k k ! Γ ( ν + k + 1 )

d d x

ν J

( x

ν ( x ))=

(5.105)

Further, as Γ ( ν + k + 1 )=( ν + k ) Γ ( ν + k ) , we obtain:

( − 1 ) k 2 k + ν − 1 k ! Γ ( ν − 1 + k + 1 ) , x 2

∞ ∑ k = 0

d d x

ν J

ν

( x

ν ( x ))= x

(5.106)

and according to (5.74)

d d x

ν J

ν J

( x

ν ( x ))= x

(5.107)

ν − 1 .