Famous Theorems of Mathematics/Euler's equation

Euler's equation states that,

From which, the famous identity,

may be deduced.

ProofEdit

Definition 1
 
Definition 2
 
Definition 3
 

The following results will be assumed;

 
 
Theorem 1
 
Proof

By definition 1,

 

Observe that one may split the summation into two,

 

Evaluating,

 
 

By definition 2 and 3,

   
Theorem 2
 
Proof

By theorem 1,

 
 

Hence,