Applied Mathematics/Complex Integration

Complex integrationEdit

On the piecewise smooth curve    , suppose the function f(z) is continuous. Then we obtain the equation below.

 

where   is the complex function, and   is the complex variable.

ProofEdit

Let

 
 

Then

 
 
 

The right side of the equation is the real integral, therefore, according to calculus, the relationship below can be applied.

 

Hence

 
 
 
 
 

This completes the proof.