Sessions

1.  CAUCHYíS INEQUALITY,  LIOUVILLEíS THEOREM, AND PARSEVALíS IDENTITY

1.1          Cauchyís Inequality

Theorem 1. (Cauchyís Inequality). Let be analytic in a disk and let be the circle . Suppose in for some positive constant . Then

                                                                                                     

 

Proof. By the previous corollary we get

                                                                                                     

 

The result follows after letting  .

 

1.2          Liouvilleís Theorem

 

Theorem 2. (Liouvilleís Theorem).  A bounded entire function must be constant.

Proof. By the previous corollary for and for every we can write . Let Then . Since was arbitrary, But then and thus  is constant.

 

1.3          Parsevalís Identity

Theorem 3. (Parsevalís Identity).  Let and for let

.

Then for , we have .

Proof.  Consider the partial sum

of the series  .

Then

Multiplying out, integrating from 0 to , and using when and when j=k we get

Since uniformly on @ and is bounded on , it follows that

uniformly on .  Letting gives Parsevalís identity.