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PHYS809 Class 17 Notes
Eigenfunction expansions for Green functions
When we considered the Green function for the interior of a rectangle, we found that there was an
asymmetry in the dependences on the coordinates. Here we briefly discuss how to develop Green
functions that put the coordinates on an equal footing. Note that this can be done only at the expense
of an additional sum over indices.
Consider the equation
2
0,
λ
∇ Φ + Φ =
(17.1)
subject to boundary conditions
0
Φ =
on
x
= 0,
x
=
a
,
y
= 0 and
y
=
b
. By the method of separation of
variables we find that nontrivial solutions are possible only if
λ
takes discrete
eigenvalues
given by
2
2
,
mn
m
n
a
b
π
=
+
(17.2)
where
m
and
n
are positive integers. The solutions are then the orthonormal
eigenfunctions
2
sin
sin
.
mn
m x
n y
a
b
ab
Φ
=
(17.3)
The complete solution is a linear combination of terms of this form.
The Green function is a solution of the Poisson equation
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 Fall '11
 MacDonald

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