1
PHYS809Class14Notes
Separation of variables in cylindrical coordinates
In cylindrical coordinates
,
,
,
r
z
θ
the metric is
2
2
2
2
2
.
ds
dr
r d
dz
θ
=
+
+
(14.1)
Hence the scale factors are
1
2
3
1,
,
1,
h
h
r h
=
=
=
so that Laplace’s equation for the potential is
2
2
2
2
2
1
1
0.
r
r
r
r
r
z
θ
∂
∂Φ
∂ Φ
∂ Φ
+
+
=
∂
∂
∂
∂
(14.2)
To separate the variables, let
( )
(
)
( )
,
R r
Z
z
θ
Φ =
Θ
so that
2
2
2
2
2
1 1
1 1
1
0.
d
dR
d
d Z
r
R r dr
dr
r
d
Z dz
θ
Θ
+
+
=
Θ
(14.3)
Assuming that
θ
is unrestricted, the angular dependence is
,
im
e
θ
±
Θ =
(14.4)
where
m
is an integer. Equation (14.3) is then
2
2
2
2
1 1
1
.
d
dR
m
d Z
r
R r dr
dr
r
Z dz

= 
(14.5)
The right hand side can be equal to a positive or negative constant. We will first consider the case in
which the constant is negative. In particular, we take
2
2
2
,
d Z
k Z
dz
=
(14.6)
which has solutions of form
.
kz
Z
e
±
=
(14.7)
This form of solution is appropriate for problems in which
z
goes to infinity.
The radial equation is now
2
2
2
1
0.
d
dR
m
r
k
R
r dr
dr
r
+

=
(14.8)
This is Bessel’s equation. By multiplying by
2
r
to get
(
)
2
2
2
0,
d
dR
r
r
k r
m
R
dr
dr
+

=
(14.9)
we see that the solution is a function of
x
=
kr
.
As for Legendre’s equation, we look for a series solution of form
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2
0
,
n
l
l
l
R
x
a x
∞
=
=
∑
(14.10)
with
0
0.
a
≠
Substitution into Bessel’s equation leads to
(
)
2
2
2
2
0
0
2
.
l
n
l
n
l
n
l
l
l
l
l
l
l
n
m
a x
a x
a
x
∞
∞
∞
+
+ +
+

=
=
=
+

= 
= 
∑
∑
∑
(14.11)
Since the lowest order term on the right hand side has degree
n
+ 2
, the coefficients of the
x
n
and
x
n
+1
terms must both be identically zero. Hence
(
)
2
2
2
2
1
0 and
1
0.
n
m
n
m
a

=
+

=
(14.12)
The first equation gives
,
n
m
= ±
and hence the second requires
1
0.
a
=
Thus all odd terms in the series in (14.10) have coefficients equal to zero.
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 Fall '11
 MacDonald
 Recurrence relation, Bessel function, Bessel Functions

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