Elasticity_ Barber

282 18 Preliminary mathematical results
A related result can be obtained for the exterior problem in which f is
holomorphic in the multiply connected region Ω exterior to the closed contour
S and extending to and including the point at infinity. In this case, a general
holomorphic function f can be expressed as a Laurent series
f = C 0 +
∞∑
k=1
C k
ζ k , (18.45)
where the origin lies within the hole excluded by S. For the exterior problem
to be well posed, we must specify the value of f at all points on S and also
at infinity ζ →∞. We shall restrict attention to the case where f(∞)=0 and
hence C 0 =0, since the more general case is most easily handled by first solving
a problem for the entire plane (with no hole) and then defining a corrective
problem for the conditions at the inner boundary S.
S 1
S 3
S
S 2
●
ζ = ζ 0
Figure 18.3: Contour for the region exterior to a hole S.
To invoke the Cauchy integral theorem, we need to construct a contour
that encloses a simply connected region including the general point ζ =ζ 0 . A
suitable contour is illustrated in Figure 18.3, where the path S corresponds
to the boundary of the hole and S 1 is a circle of sufficiently large radius to
include the point ζ =ζ 0 . Notice that the theorem requires that the contour be
traversed in an anticlockwise direction as shown.
Equation (18.44) now gives
f(ζ 0 ) = 1
2πı
∮
S+S 1+S 2+S 3
f(s)ds
(s − ζ 0 ) . (18.46)
In this integral, the contributions from the line segments S 2 , S 3 clearly cancel,
leaving
f(ζ 0 ) = − 1 ∮
f(s)ds
2πı S (s − ζ 0 ) + 1 ∮
f(s)ds
2πı S 1
(s − ζ 0 ) . (18.47)