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
(
)
.

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- Fall '11
- MacDonald
- Coordinate system, Polar coordinate system, Dirac delta function, Cylindrical coordinate system, cylindrical polar coordinates
-
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