Step 1

Cauchy’s Inequality

We state and prove an inequality due to Cauchy which has numerous applications including proofs of many theorems.

Step 2

Liouville’s Theorem

We now state and prove a theorem due to Cauchy which gives the simplest proof of the Fundamental Theorem of Algebra.

Step 3

Parseval’s Identity

We conclude this session with an extremely important identity from which Cauchy’s Inequality, Liouville’s Theorem, and the Maximum principle all follow as simple corollaries.  This identity is most famous for finding exact values of many numerical series in connection with Fourier Theory.