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.