
17.5. Construction of the Green’s Functions. General Formulas and Relations 1229
3
◦
. Third boundary value problem. The boundary condition:
∂
z
w − kw = f(x) at z = 0.
Green’s function:
G(x, y, z, ζ) = E
e
(x, y, z − ζ) + E
e
(x, y, z + ζ) −2k
Z
∞
0
e
−ks
E
e
(x, y , z + ζ + s) ds
= E
e
(x, y, z − ζ) + E
e
(x, y, z + ζ) −2k
Z
∞
z+ζ
e
−k(σ−z−ζ)
E
e
(x, y , σ) dσ.
◮ Domain: x ∈ R
n
, 0 ≤ z ≤ l. Problems for elliptic equations.
1
◦
. First boundary value problem. Boundary conditions:
w = f
1
(x) at z = 0, w = f
2
(x) at z = l.
Green’s function:
G(x, y, z, ζ) =
∞
X
n=−∞
E
e
(x, y, z − ζ + 2nl) −E
e
(x, y, z + ζ + 2nl)
. (17.5.3.3)
Domain of the free parameters: y ∈ R
n
and 0 ≤ ζ ≤ l.
2
◦
. Second boundary ...