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. |
