Class_16new - PHYS809 Class 20 Notes Green function in...

1
PHYS809Class20Notes
Green function in cylindrical polar coordinates
In cylindrical polar coordinates
( )
, , , z ρ φ
the Green function is a solution of
( ) ( ) ( ) ( )
2
4
, , , , , . G z z z z
π
ρ φ ρ φ δ ρ ρ δ φ φ δ
ρ
′ ′ ′ ′ ′ ′
∇ = - - - - (16.1)
If
φ
and
z
are unrestricted, we can represent the last pair of Dirac delta functions by
( ) ( ) ( ) ( ) ( )
0
1 1 1
, cos .
2 2
im ik z z
m
e z z e dk k z z dk
φ φ
δ φ φ δ
π π π
∞ ∞
∞
′ ′ - -
=-∞
-∞
′ ′ ′
  - = - = = -
 
∑
∫ ∫
(16.2)
We then represent the Green function in a similar way
( )
( )
( ) ( )
2
0
1
, , , , , cos , , .
2
im
m
m
G z z e k z z g k dk
φ φ
ρ φ ρ φ ρ ρ
π
∞
∞
′ -
=-∞
′ ′ ′ ′ ′  
= -
 
∑
∫
(16.3)
Substitution into equation (16.1) gives that
( )
, ,
m
g k ρ ρ
′
is a solution of
( )
2
2
2
1 4
. m
m
g m
k g
π
ρ δ ρ ρ
ρ ρ ρ ρ ρ
 
  ∂ ∂
′
- + = - -
   
∂ ∂
 
 
(16.4)
The solutions to the homogeneous equations are the modified Bessel functions
( )
m
I k ρ
and
( )
.