Mathematical Physics Vol 1

Chapter 5. Series Solutions of Differential Equations. Special functions

238

Lest us first find y 1 , assuming that x > 0, for λ 1 = ν

∞ ∑ k = 0

ν

k , a

y 1 = x

a k x

0̸ = 0 ,

(5.68)

where this series is convergent for each finite x . As in the previous case, from the condition

∞ ∑ k = 2

− 2

ν + 1

ν

( k + ν ) 2 a

x k = 0 ,

L ( y 1 )=( ν + 1 ) a 1 x

+ x

k + a k

(5.69)

we can obtain

a 2 s + 1 = 0 , s = 0 , 1 , 2 ,... a 2 s = ( − 1 ) s a 0

(5.70)

2 2 s s ! ( ν + 1 )( ν + 2 ) ··· ( ν + s ) .

The particular solution has the form

( − 1 ) s s ! ( ν + 1 )( ν + 2 ) ··· ( ν + s ) x 2 2 s .

∞ ∑ s = 1

y 1 ( x )= a 0 x ν

ν

+ a 0 x

(5.71)

It is convenient to choose the following value for a 0 (in order to connect the above relation to another special function) a 0 = 1 2 ν Γ ( ν + 1 ) , (5.72) where Γ is the so called gamma function. 8 For a 0 chosen in this way y 1 becomes y 1 = x 2 ν ∞ ∑ s = 0 ( − 1 ) s s ! Γ ( s + ν + 1 ) x 2 2 s . (5.73)

Definition The function J ν ( x ) , defined by the relation J ν = x 2 ν ∞ ∑ s = 0

( − 1 ) s s ! Γ ( s + ν + 1 ) x

2

2 s

(5.74)

.

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

As λ 1 − λ 2 is not an integer, we will search for the second particular solution in the following form y 2 = x λ 2 ∞ ∑ k = 0 b k x k = x − ν ∞ ∑ k = 0 b k x k . (5.75) 8 The Γ function will be discussed later in more detail. At this point we shall only state its definition and some of its properties, to make the text easier to follow.

∞ Z 0

x ν − 1 e − x d x

Γ ( ν )=

, for ν > 0 .

For the function defined in this way the following stands

Γ ( ν + 1 )= ν Γ ( ν ) , Γ ( 1 )= 1 , Γ ( 1 / 2 )= √ π .