Special Functions - Heat Conduction: Mathematical Models and Analytical Solutions - page 429

For c = − γ , similarly, we obtain another particular solution of Eq. (A.10)

x
2

− γ + 2 m + ∞

m = 0

m

( − 1 )

J − γ ( x )= . (A.13)

m ! Γ ( − γ + m + 1 )

When γ = n = 0 , 1 , 2 , ··· , in particular,

x
2

n + 2 m + ∞

m = 0

m

( − 1 )

J n ( x )= ,

m ! ( n + m ) !

and consequently,

x 2
2 2 +

x 4 x 2 k

2 2 k

k

J 0 ( x )= 1 −

2 −··· +( −

1 )

2 + ··· ,

2 4

( 2! ) ( k ! )

x 3
2 3 2! +

x 5
2 5 2!3! −··· +( −

x 2 k + 1
2 2 k + 1 k !

x
2 −

k

! + ··· .
By the last remark in the present appendix, for any natural number N ,
1

J 1 ( x )= 1 )

( k + 1 )

) =

0 , m = 0 , 1 , 2 , ··· , N − 1 .

Γ ( − N + m + 1

Thus

x

2 − N + 2 m

+ ∞
m = N ( − 1 )

1

m

J − N ( x )=

m ! Γ ( − N + m + 1 )

N x N
2 N N ! −

x N + 2
2 N + 2

x N + 4

=( − 1 )

! + !2! + ···

2 N + 4

( N + 1 ) ( N + 2 )

N J N (

=( − 1 ) x ) .

Therefore, J n ( x ) and J − n ( x ) are linearly dependent for any natural number n .When γ

is a positive but not natural number, J γ ( 0 )= 0and J − γ ( 0 )= ∞ . Therefore, J γ ( x ) and

J − γ ( x ) are linearly independent. Thus we obtain the general solution of Eq. (A.10)

y = C 1 J γ ( x )+ C 2 J − γ ( x ) . (A.14)

By letting C 1 = cot γπ and C 2 = − csc γπ in Eq. (A.14), we obtain another solution

J γ ( x ) cos γπ − J − γ ( x )

Y γ ( x )= . (A.15)

sin γπ

It is called the Bessel function of γ -th order of the second kind.

Since J γ ( 0 )= 0and Y γ ( 0 )= ∞ , Y γ ( x ) and J γ ( x ) are linearly independent. When

γ is not an integer, the general solution of Eq. (A.10) can also be written as

y = C 1 J γ ( x )+ C 2 Y γ ( x ) .

n (0 or natural numbers), the right-hand side of Eq. (A.15) becomes 0

When γ =

0 .