|
(1) |
|
(2) |
|
(3) |
Now divide by , |
(4) |
|
(5) |
The solution to the second part of (5) must be sinusoidal, so the differential equation is |
(6) |
which has solutions which may be defined either as a complex function with , ..., |
(7) |
or as a sum of real sine and cosine functions with , ..., |
(8) |
Plugging (6) back into (7), |
(9) |
The radial part must be equal to a constant
|
(10) |
|
(11) |
|
(12) |
Then
|
(13) |
|
(14) |
|
(15) |
This must hold true for all powers of . For the term (with ), |
(16) |
which is true only if and all other terms vanish. So for , . Therefore, the solution of the component is given by |
(17) |
Plugging (17) back into (◇), |
(18) |
|
(19) |
|
(20) |
where
|
(21) |
|
(22) |
Some of the normalization constants of can be absorbed by and , so this equation may appear in the form |
(23) |
where
|
(24) |
|
(25) |
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