9. CAUCHY’S THEORY
9.1 Cauchy’s Theorem for a Triangle
Theorem 1.
(Cauchy’s theorem for a triangle). Let
be analytic in a domain
and let
be continuous on
If C is a triangle in
, then
Proof.
By Green’s Theorem (exercise 1):
where
is the inside of the triangle C
Put
Since
is analytic on
,
and
satisfy the
Cauchy-Riemann
equations and thus
Remarks.
Indeed, the hypothesis that
is continuous on
is not needed and there are other
technical proofs. The hypotheses in Green’s Theorem are
satisfied under the assumption that
is continuous on
9.2 Cauchy’s Theorem for a
star-like domain
Theorem 2.
(Cauchy’s theorem). If
is analytic in a star-like domain
and is a closed contour in
, then
Proof.
Since
is a star-like domain, there is
such that the segment between
and
is in
for each
Define
where the path of integration is that
segment. Let
be a point in
off this line segment.
By continuity of
at
, given
we choose
small enough so that
Now let
be small enough and satisfying
Consider the triangle with vertices
Then, by Cauchy’s Theorem for a
triangle,
Hence
Then
It follows that
exists and is equal to
Now with
denoting C we have
Thus
since C is closed.
9.3 Cauchy’s Integral Theorem
Theorem 3.
(Cauchy’s Integral Theorem). Let
be analytic in a disk
and C is the circle
. Then for every
in the disk
we have
Proof.
Consider the function
if
and
if
.
Clearly, g is analytic in the star-like domain
and by hypothesis C is inside
. Thus by Cauchy’s Theorem
; that is,
or
and the result follows.
9.4 Cauchy’s Integral Formula
Theorem 4.
Proof.
Differentiate Cauchy’s Integral Formula n times
with respect to z. |