Colloquium publications.

98

THE CAMBRIDGE COLLOQUIUM.

58. The Green's Function. The Green's function for (41) is
closely related to the function k(xy x'y'). As a function of
(x'y') it is a solution of (41) itself, with f = 0, and it may be
deduced from this fact, and from (47), that the Green's function
satisfies the equation
g(xy x'y') = G(xy xy') + 2- ff G( x 'y')ag(xy[ )dd,
and hence the integral equation
dx' g(xy I x'y') = Ox'y'G(xy I x'y')
+ f2 f- x Gl x'yy')atg(xy ^ q.
XRXf
Hence we have
(50) Odxg(xy xy') = - ( 2 I_ - axyG(xy xy'),
and
g(xy x'y') = G(xy x'y')
(51) - ffG(I |xi ) (2 tG(xy I t)) dld.
~ 4. THE PARABOLIC INTEGRO-DIFFERENTIAL EQUATION OF THE
USUAL TYPE
59. The Generalized Green's Function. For simplicity, let us add to the
conditions on the boundary of the region, the requirements 42'(x) > 0,
41'(x) < 0, and deal with the equation
au a2U,, C,.a-_u _ 2 = jf A(x, r, y)u(, y)dL,
or for still greater simplicity, with its generalization:
(52) f udy - ) dx = ff da f A (x y)u(xy)dx.
In these equations (xy, y) denotes the point on the boundary xorx of which one
co-ordinate is y.
If u(xy) and v(xy) are any two continuous functions, it may be proved by
means of a change in the order of integration that the following identity