BESSEL FUNCTIONS By Tom Irvine Email: March 1, 2012

BESSEL FUNCTIONS
By Tom Irvine
Email: [email protected]
March 1, 2012
______________________________________________________________________________
Differential Equation
x 2 y  xy  (x 2  n 2 ) y  0
Solution
y(x)  A1J n (x)  A 2 Yn (x)
y(x)  A1J n (x)  A 2 J  n (x)
for all n
for all n≠ 0, 1, 2, …
Bessel Function of the First Kind
Zero Order
J 0 (x)  1 
x 22  x 24  x 26  
1!2 2!2 3!2
First Order
J1 ( x ) 

x  x 22 x 24 x 26
d


    J 0 ( x )
1 
2
dx

21!2 32!2 43!2

1
Order n
n = 0, 1, 2, …
J n (x) 
 (1) k ( x / 2 ) 2k  n

k 0
J  n (x) 
k! (k  1  n )
 (1) k ( x / 2 ) 2k  n

k 0
k! (k  1  n )
  1n J n ( x )
Modified Bessel Function of the First Kind
Zero Order
I 0 (x)  1 
x 22  x 24  x 26  
1!2 2!2 3!2
First Order
I1 ( x ) 
 d
x  x 22 x 24 x 26
I 0 (x)


  
1 
2
2
2
2
dx







2
1
!
3
2
!
4
3
!


Higher Order Modified Bessel Function
I(x)   j  n J n ( j x)
2
Bessel Function of the Second Kind (Neumann Functions)
Zero Order
Yo ( x ) 

2  x
2
2
2

 ln  c J o ( x )  J 2 ( x )  J 4 ( x )  J 6 ( x )  

  2
1
2
3


where c = 0.577 215 665
First Order
Y1 ( x ) 

2  x
9
d

1 1
 ln  c J1 ( x )     J1 ( x )  J 3 ( x )     Yo ( x )

  2
4
dx

x 2

Modified Bessel Functions of the Second Kind
Zero Order
 x

2
2
2

K o ( x )    ln  c  I o ( x )  I 2 ( x )  I 4 ( x )  J 6 ( x )  
1
2
3

 2

Half-Odd Integers
J 1 / 2(x) 
2
sin( x )
x
J 3 / 2(x) 
2  sin( x )

 cos( x )

x  x

Y 1 / 2(x)  
Y 3 / 2(x)  
2
cos(x )
x
2  cos( x )

 sin( x )

x  x

3
Recurrence Relations
J n 1 ( x ) 
2n
J n ( x )  J n 1 ( x )
x
I n 1 ( x )  
2n
I n ( x )  I n 1 ( x )
x
4
APPENDIX A
Bessel Function of the First Kind of Order n
First Derivative
d
n
J n ( x )  J n 1 ( x )  J n ( x )
dx
x
n
 J n 1 ( x )  J n ( x )
x
1
 J n 1 ( x )  J n 1 ( x )
2
Second Derivative
d2
dx 2
J n (x)

1 d
J n 1 (x )  J n 1 (x )
2 dx

1 1
1

 J n  2 ( x )  J n ( x )  J n ( x )  J n  2 ( x )
2 2
2


1
 J n  2 ( x )  2J n ( x )  J n  2 ( x )
4
5
Third Derivative
d3
dx 3
J n (x)

1 d
 J n  2 ( x )  2J n ( x )  J n  2 ( x )
4 dx

1 d
d
d

J n  2 (x)  2 J n (x) 
J n  2 ( x )

4  dx
dx
dx


1 1
1

 J n  3 ( x )  J n 1 ( x )  J n 1 ( x )  J n 1 ( x )  J n 1 ( x )  J n  3 ( x )
4 2
2


1
J n  3 ( x )  J n 1 ( x )  2J n 1 ( x )  J n 1 ( x )  J n 1 ( x )  J n  3 ( x ) 
8

1
J n  3 ( x )  3J n 1 ( x )  3J n 1 ( x )  J n  3 ( x )
8
6
Fourth Derivative
d4
dx 4
J n (x)

1 d
J n  3 ( x )  3J n 1 ( x )  3J n 1 ( x )  J n  3 ( x )
8 dx
1 d
d
d
d

  J n  3 ( x )  3 J n 1 ( x )  3 J n 1 ( x ) 
J n  3 ( x )
8  dx
dx
dx
dx

1 d
d
d
d

  J n  3 ( x )  3 J n 1 ( x )  3 J n 1 ( x ) 
J n  3 ( x )
8  dx
dx
dx
dx


1
J n  4 ( x )  J n  2 ( x )
16

3
J n  2 ( x )  J n ( x )
16

3
J n ( x )  J n  2 ( x )
16

1
J n  2 ( x )  J n  4 ( x )
16

1
J n  4 ( x )  4J n  2 ( x )  6J n ( x )  4J n  2 ( x )  J n  4 ( x )
16
7
APPENDIX B
Modified Bessel Function of the First Kind of Order n
First Derivative
d
n
I n ( x )  I n 1 ( x )  I n ( x )
dx
x
n
 I n 1 ( x )  I n ( x )
x
1
 I n 1 ( x )  I n 1 ( x )
2
Second Derivative
d2
dx 2
I n (x) 
1 d
I n 1 (x )  I n 1 ( x )
2 dx

1 d
d

I n 1 ( x ) 
I n 1 ( x )

2  dx
dx


1 1
1

 I n  2 ( x )  I n ( x )  I 0 ( x )  I n  2 ( x )
2 2
2


1
I n  2 ( x )  2I n ( x )  I n  2 ( x )
4
8
Third Derivative
d3
dx 3
I n (x) 
1 d
I n  2 ( x )  2I n ( x )  I n  2 ( x )
4 dx

1 d
d
d

I n  2 ( x )
 I n  2 (x)  2 I n (x) 
4  dx
dx
dx


1 1
1

 I n  3 ( x )  I n 1 ( x )  I n 1 ( x )  I n 1 ( x )  I n 1 ( x )  I n  3 ( x )
4 2
2


1
I n  3 ( x )  I n 1 ( x )  2I n 1 ( x )  I n 1 ( x )  I n 1 ( x )  I n  3 ( x )
8

1
I n  3 ( x )  3I n 1 ( x )  3I n 1 (x )  I n  3 ( x )
8
9
Fourth Derivative
d4
dx 4
I n (x)

1 d
I n  3 ( x )  3I n 1 ( x )  3I n 1 ( x )  I n  3 ( x )
8 dx
1 d
d
d
d

  I n  3 ( x )  3 I n 1 ( x )  3 I n 1 ( x ) 
I n  3 ( x )
8  dx
dx
dx
dx

1 d
d
d
d

  I n  3 ( x )  3 I n 1 ( x )  3 I n 1 ( x ) 
I n  3 ( x )
8  dx
dx
dx
dx


1
I n  4 ( x )  I n  2 ( x )
16

3
I n  2 ( x )  I n ( x )
16

3
I n ( x )  I n  2 ( x )
16

1
I n  2 ( x )  I n  4 ( x )
16

1
I n  4 ( x )  4I n  2 ( x )  6I n ( x )  4I n  2 ( x )  I n  4 ( x )
16
10
APPENDIX C
Bessel Function of the First Kind of Order Zero, Derivatives
First Derivative
d
J 0 ( x )  J1 ( x )
dx
Second Derivative
d2
dx 2
J 0 (x)

1
 J  2 (x)  2J 0 (x)  J 2 (x)
4
Note that
J n 1 (x)  J n 1 (x) for n=0
Thus
d2
dx 2
d2
dx 2
1
  J 0 (x)  2J 0 (x)  J 0 (x)
4
J 0 (x)

J 0 (x)
 J 0 ( x )
11
Third Derivative
d3
dx 3
J 0 (x)

1
J 3 ( x )  3J 1 ( x )  3J1 ( x )  J 3 ( x )
8

1
3J1 ( x )  J 3 ( x )
4

1
3J1 ( x )  J1 ( x )
4
 J1 ( x )
Fourth Derivative
d4
J 0 (x)

1
J  4 (x)  4J  2 (x)  6J 0 (x)  4J 2 (x)  J 4 (x)
16
J 0 (x)

1
 J  2 (x)  4J  2 (x)  6J 0 (x)  4J 2 (x)  J 2 (x)
16
J 0 (x)
dx 4

1
 5J  2 (x)  6J 0 (x)  5J 2 (x)
16
J 0 (x)

1
5J 0 (x)  6J 0 (x)  5J 0 (x)
16
J 0 (x)
 J 0 (x)
dx 4
d4
dx 4
d4
d4
dx 4
d4
dx 4
12
APPENDIX D
Modified Bessel Function of the First Kind of Order Zero, Derivatives
First Derivative
d
1
I n ( x )  I 1 ( x )  I1 ( x )
dx
2
 I1 ( x )
Second Derivative
d2
I 0 (x)
dx 2

1
 I  2 (x)  2I 0 (x)  I 2 (x)
4
Note that
I n 1 (x)  I n 1 (x) for n=0
Thus
d2
I 0 (x)
2
dx
d2
dx 2
I 0 (x)

1
 I 0 (x)  2I 0 (x)  I 0 (x)
4
 I 0 (x)
13
Third Derivative
d3
dx 3
I 0 (x)

1
I 3 ( x )  3I 1 ( x )  3I1 ( x )  I 3 ( x )
8

1
3I1 ( x )  I 3 ( x )
4

1
3I1 ( x )  I1 ( x )
4
 I1 ( x )
Fourth Derivative
d4
I 0 (x)

1
I  4 (x)  4I  2 (x)  6I 0 (x)  4I 2 (x)  I 4 (x)
16
I 0 (x)

1
I  2 (x)  4I  2 (x)  6I 0 (x)  4I 2 (x)  I 2 (x)
16
I 0 (x)
dx 4

1
5I  2 (x)  6I 0 (x)  5I 2 (x)
16
I 0 (x)

1
5I 0 (x)  6I 0 (x)  5I 0 (x)
16
I 0 (x)
 I 0 (x)
dx 4
d4
dx 4
d4
d4
dx 4
d4
dx 4
14