182 3 Real Analysis
As a corollary, we obtain the well-known fact that a container filled with gas under
pressure is at equilibrium; a balloon will never move as a result of internal pressure.
We conclude our series of examples with an application of Green’s theorem, the
proof given by D. Pompeiu to Cauchy’s formula for holomorphic functions. First, let
us introduce some notation for functions of a complex variablef(z)=f(x+iy)=
u(x, y)+iv(x, y).Ifuandvare continuously differentiable, define
∂f
∂z=
1
2
[
∂f
∂x+i
∂f
∂y]
=
1
2
[(
∂u
∂x−
∂v
∂y)
+i(
∂u
∂y+
∂v
∂x)]
.
The functionfis called holomorphic if∂f∂z=0. Examples are polynomials inzand any
absolutely convergent power series inz. Also, letdz=dx+idy.
Cauchy’s theorem.Letbe an oriented curve that bounds a region on its left, and
leta∈ .Iff(z)=f(x+iy)=u(x, y)+iv(x, y)is a holomorphic function on
such thatuandvare continuous on ∪and continuously differentiable on , then
f(a)=1
2 πi∮
f(z)
z−adz.Proof.The proof is based on Green’s formula, applied on the domain (^) obtained from
by removing a disk of radiusaroundaas described in Figure 23 toP =Fand
Q=iF, whereFis a holomorphic function to be specified later. Note that the boundary
of the domain consists of two curves,and.
∆ε Γ
a
Γε
Figure 23
Green’s formula reads
∮
Fdz−
∮
Fdz=∮
Fdx+iFdy−∮
Fdx+iFdy=
∫∫
(^)
i
(
∂F
∂x+i∂F
∂y)
dxdy= 2 i∫∫
(^)
∂F
∂zdxdy= 0.Therefore,
∮
F(z)dz=∮
F(z)dz.