
22.3. Analytical Solutions of Mathematical Problems 1307
Here X = (x, y), X
0
= (x
0
,y
0
), the function v(X, X
0
) is the free space Green’s function (does not
depend o n the bo undary conditions), and the function w(X ,X
0
) satisfies the Laplace equation and the
boundary con ditions (and is regular at X = X
0
); i.e., ∇
2
w(X,X
0
) = 0 in V and w(X, X
0
) = −v (X ,X
0
)
(i.e., G(X ,X
0
) = 0) on S for the Dirichlet boundary cond itions.
It is well known that the 2D free space function v(X, X
0
) is
v(X,X
0
) = −
1
4π
ln
(x −x
0
)
2
+ (y −y
0
)
2
.
If to v(X,X
0
) we add the function
w(X,X
0
) =
1
4π
ln
(x −x
0
)
2
+ (y + y
0
)
2
,
which satisfies the Laplace equa tion ∇
2
w(X,X
0
) = 0 in V and is regular ...