#
1-
Morera’s Theorem
The following theorem can be considered a converse of
Cauchy’s Theorem and is often useful in proving that a
function is analytic:
## Theorem (Morera).
Suppose
that *f(z)* is continuous in a domain D and
for every triangle *T* which together with its
interior lie in *D.* Then *f(z) *is analytic in
*D.*
To prove Morera’s Theorem we use the following lemma whose
proof is a slight modification of the proof of Cauchy’s
Theorem:
##
##
Lemma.
Suppose
that *f(z)* is continuous in a star domain *D*
and
for every triangle *T* which together with its
interior lie in *D.* Then there exists a function *
F(z) * analytic in *D *such that* F’(z)=f(z).*
**
**
**
Proof of Morera’s Theorem. **Let
be any point in *D.
*Choose a positive number
*r *small enough so that the disk
Thus the star
domain
satisfies the hypothesis of the lemma and so there
is an analytic function *F(z)* such that* F’(z)=f(z)
*in
. Since
*F(z)* is analytic in
it has
analytic derivatives of all orders in
and, in
particular, *F’(z)=f(z)* is analytic* *in
. But
was an arbitrary point in *D.
*Consequently, *f(z)*
is analytic in *D.*
*
*
**
Corollary. **Suppose
that *f(z)* is continuous in a domain D and analytic
at all points of *D *except possibly at the points of
some line
*.* Then *f(z) *is analytic on *
D.*
**
Proof. **Rotating if necessary, we may assume that
is parallel to
the *x-*axis. Now the result follows from Morera’s
Theorem.
*
*
#
2- Analytic Continuation Principle
We state without proof the following useful:
**
**
**
Theorem (Analytic Continuation Principle). **If two analytic functions *f* and *g* agree in a
neighborhood of a point of a domain *D, *then they
agree on *D.*
**
**
#
3- Reflection Principle
**
Theorem. **Let
*D* be a domain of
C
symmetric with respect to the real axis. Let
,
,
and
.
Let *g*(*z*) be continuous on
, holomorphic on
*G* and
R.
Then there is a unique function
holomorphic on *D*
such that its restriction to
is *g*(*z*).
**
Proof. **Consider the function *h*(*z*) defined on
by
. Since the function
is continuous, *h*(*z*)
is continuous on
. We now prove that
*h*(*z*)
is holomorphic on *L* by proving that it is
holomorphic at each
Now
. Since *g*(*z*) is
holomorphic on *G*, it is analytic there and so there
exists* r>0 *such that
and
.
Now for all
we
have
.
So by a limiting process we get
and therefore *h*(*z*) is analytic on *L*
and hence holomorphic there. Consider the function
on
*D* defined by
=* g*(*z*) if
and *h*(*z*) if
. By
hypothesis, *g*(*z*) is real on
. Thus
. So
is continuous on *D* and holomorphic on *D*
except possibly on
R.
Thus,
is holomorphic on *D *by the previous
corollary. Clearly,
extends *g*(*z*) and
is
unique by the Analytic Continuation Principle.
**
Example. **Suppose that *f(z) *is an entire function that is
real-valued on an interval (*a,b*) of the real axis.
Put
if
and
if
. By the reflection principle *F*(*z*)
is analytic on the union *D *of the upper and lower
semi-planes and (*a,b*). Moreover, *f(z) *is
analytic on *D* and agrees with *F*(*z*)
for. Thus,
on *D.
*In particular,. Furthermore,
*f*(*z*) is continuous on
C
. Consequently, for any
R,
and therefore *f*(*z*) is real on the real axis. |