SPECIAL FUNCTIONS - Water Quality Engineering in Natural Systems - page 417

The Bessel function J − n ( x ) can be derived from Equa-
tion (D.14) by simply replacing n by − n in the formula.
A convenient relationship to note is

d y
dx

2

x dy
dx

x

2

+ + ( x

2

− n y

2

) = 0 , n ≥ 0 (D.7)

2

is called Bessel's equation . The general solutions of Bes-
sel's equation are

J ( ) x = − ( 1 )

n

J ( ), x n = 0 1 2 … , , , (D.15)

− n n

y AJ x BJ = ( ) + ( ), x n ≠ 1 2 … , ,

(D.8)

n − n

D.2.2.2 Bessel Function of the Second Kind of Order n.
This function is given by

y AJ x BY x = ( ) + ( ), all n

(D.9)

n n

where J n ( x ) is called the Bessel function of the first kind
of order n , and Y n ( x ) is called the Bessel function of the
second kind of order n .

J x ( )cos n π

π

− J ( ) x

n − n

, n ≠ 0 1 2 … , , ,

sin n

Y x ( ) =

If Bessel's equation (Eq. D.7) is slightly modified and
written in the form

n

J x ( )cos p π − J ( ) x

p − p

lim , n = 0 1 2 … , , ,

sin p π

p n →

(D.16)

d y
dx

2

x dy
dx

x

2

+ − ( x

2

+ n y

2

) = 0 , n ≥ 0 (D.10)

2

D.2.2.3 Modiied Bessel Function of the First Kind
of Order n. This function is given by

then this equation is called the modified Bessel's equa-
tion . The general solutions of the modified Bessels equa-
tion are

I x ( )

n

y AI x BI = ( ) + ( ), x n ≠ 1 2 … , ,

(D.11)

n − n

x
n

n

x
n

2

x

4

= 1 + + +

y AI x BK x = ( ) + ( ), all n

(D.12)

2

n

Γ ( + 1 ) 2 2 ( + 2 ) 2 4 2 ⋅ ( n + 2 2 )( n + 4 )

n n

(D.17)

where I n ( x ) is called the modified Bessel function of the
first kind of order n , and K n ( x ) is called the modified
Bessel function of the second kind of order n .

∞
∑ ( / )

x 2

n + 2 k

= (D.18)

k ! ( Γ n k + + 1 )

k = 0

D.2.2 Evaluation of Bessel Functions

The Bessel function I − n ( x ) can be derived from Equa-
tion (D.18) by simply replacing n by − n in the formula.
A convenient relationship to note is

The solution of Bessel's equations can be found in
most Calculus texts, for example, Hildebrand (1976).
The Bessel functions cannot generally be expressed
in closed form, and are usually presented as infinite
series.

I ( ) x = I x n ( ), = 0 1 2 … , , , (D.19)

− n n

D.2.2.1 Bessel Function of the First Kind of Order n.
This function is given by

D.2.2.4 Modified Bessel Function of the Second Kind
of Order n. This function is given by

J x ( )

n

n 2 4

x
n

x
n

x

= 1 − + −

2

n

Γ ( + 1 ) 2 2 ( + 2 ) 2 4 2 ⋅ ( n + 2 2 )( n + 4 )

π

[ I ( ) x − I x ( )], n ≠ 0 1 2 , , ,

…

− n n

2 sin n π

π

(D.13)

K x ( ) =

n

[ I ( ) x − I x

p ( )],

n = 0 1 2 … , , ,

lim

− p

2 sin p π

∞
∑ (

− ) ( / )

! (

1

k

x 2

n + 2 k

p n →

=

(D.14)

k Γ n k + + 1 )

(D.20)

k = 0