Applied Mathematics/Complex Integration

Complex integration

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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.

Proof

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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.