Mathematical Physics Vol 1

Chapter 5. Series Solutions of Differential Equations. Special functions

240

Definition The function K n defined by the relation

1 2 1 2 1 2

x 2

j !

x 2

− n n − 1 ∑ j = 0

( n − j − 1 ) !

2 j

K n ( x )= −

−

1 n

1 n !

x 2

1 2

n + J n ( x ) log x −

1 +

−

+ ··· +

(5.84)

x 2

n ∞

( − 1 ) i i ! ( i + n ) ! ×

∑ i = 1

−

× 1 +

1 i

+ 1 +

1 i + n

x 2

1 2

1 2

2 i

+ ··· +

+ ··· +

.

is called the Bessel function of the second type of order n .

Some Bessel functions Let us now write the expressions for some Bessel functions

x 2 2 2

x 4 2 2 · 4 2 −

x 6 2 2 · 4 2 · 6 2

J 0 ( x )= 1 −

(5.85)

+

+ ...

x 2

+ ...

x 2 2 · 2 2

x 4 2 · 4 2 · 6 −

x 6 2 · 4 2 · 6 2 · 8

J 1 ( x )=

1 −

(5.86)

+

Let us determine the functions J n + 1 / 2 , where n is an integer. First, from (5.74), we determine J 1 / 2 and J − 1 / 2

1 2

( − 1 ) k k ! Γ ( 3

x 2

2 k +

∞ ∑ k = 0

J 1 / 2 ( x )=

(5.87)

.

2 + k )

Further, as according to (5.126) and (5.138)

1 · 3 · 5 ... ( 2 k + 1 ) 2 k + 1

1 · 3 · 5 ... ( 2 k + 1 ) 2 k + 1

√ π ,

Γ ( 3

Γ ( 1

(5.88)

2 + k )=

2 )=

by substituting into (5.87) we obtain

J 1 / 2 ( x )= r

∞ ∑ k = 0

( − 1 ) k · x 2 k + 1 ( 2 k + 1 ) ! .

2 π · x

(5.89)

As this sum represents the expansion of the function sin x into a series, we finally obtain J 1 / 2 ( x )= r 2 π · x sin x . (5.90) In a similar way, for J − 1 / 2 we obtain

x )= r

2 π · x

J

cos x .

(5.91)

− 1 / 2 (