3.4 Trefftz Integral
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3.4 Trefftz Integral
Let us consider some function harmonic in D D 0 satisfying the condition
∂∂ 0
∂z
= .
(3.12)
The condition (3.12) in D is also satisfied by the sum
= 0 + C(x, y),
(3.13)
where C(x, y) is the arbitrary function harmonic in D of two variables x and y.
We notice that
≡
x
∂∂
∂z
= 2
∂ 2
∂x∂z
=
z
∂∂
∂x
, ((( = 0),
(3.14)
we can write as
xx = z
∂∂
∂x
+ f 1 ,
where f 1 is the function harmonic in D.
Taking into account this dependency, the first of formulas (3.10) can be written
as
u = f x + f 1 −
z
2(1 − 2ν)
∂∂ 0
∂x
.
(3.15)
By expressing 0 from formula (3.13) and substituting this expression in the
dependency (3.15), we obtain
u = ϕ x −
z
2(1 − 2ν)
∂∂ 0
∂x
(3.16)
that designates
ϕ x = f x + f 1 −
z
2(1 − 2ν)
·
∂C(x, y)
∂x
.
Since f x , f 1 and
z∂C(x, y)
∂x
are functions harmonic in D, the function ϕ x is also
a function harmonic in D. Consequently, the total integral of the equation system
(3.6) can be represented as