Cauchy's integral theorem

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the Cauchy integral theorem (also known as the Cauchy–Goursat theorem) in <a href="/facts/Complex_analysis/hgCoQlH0">complex analysis</a>, named after <a href="/facts/Augustin-Louis_Cauchy/aoTjgj6j">Augustin-Louis Cauchy</a> (and <a href="/facts/%25C3%2589douard_Goursat/kMndmd26">Édouard Goursat</a>), is an important statement about <a href="/facts/Line_integral/10s4B3Nn">line integrals</a> for <a href="/facts/Holomorphic_function/kVzthtEf">holomorphic functions</a> in the <a href="/facts/Complex_number/2w7mqI7e">complex plane</a>. Essentially, it says that if 
  
    
      
        f
        (
        z
        )
      
    
    {\displaystyle f(z)}
  
 is holomorphic in a <a href="/facts/Simply_connected/xFlta63J">simply connected</a> <a href="/facts/Domain_(mathematical_analysis)/xT44oI3E">domain</a> Ω, then for any simply closed contour 
  
    
      
        C
      
    
    {\displaystyle C}
  
 in Ω, that contour integral is zero. 
</p><p>
  
    
      
        
          ∫
          
            C
          
        
        f
        (
        z
        )
        
        d
        z
        =
        0.
      
    
    {\displaystyle \int _{C}f(z)\,dz=0.}

</p>

Cauchy's integral theorem open-in-new

Cauchy's integral theorem