
836 SECOND-ORDER ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES
where
D
2n
= 2
ce
2n
(0, −q)
ce
2n
(0, q)
A
(2n)
0
, D
2n+1
= −
1
2
ce
2n+1
(0, −q)
ce
2n+1
(0, q)
αB
(2n+1)
1
, q = −
1
16
α
2
.
Here, the A
(2n)
0
and B
(2n+1)
1
are the coefficients in the series expansions of the Mathieu
functions; these can be found in M cL achlan (1947).
4
◦
. Consider the second boundary value problem in the upper half-plane (−∞ < x < ∞,
0 ≤ y < ∞). We assume that a thermal flux is prescribed on the surface of a plate of finite
length and the medium has a constant temperature far away from the plate:
∂
y
w = f (x) for y = 0, |x| < 1,
∂
y
w = 0 for y = 0, |x| > 1,
w → w
∞
as x
2
+ y
2
→ ∞.
The solution of this problem