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I am studying the book "Complex variables and applications" by James Ward Brown, Ruel Vance Churchill and I don't understand the proof given in the text.

The proof uses a theorm:

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Proof:

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How can we apply the theorm ? isn't $-C_1$ and $C_2$ in different orientation (in terms of clockwise direction) so we can't apply the theorm ?

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By definition, the line integral changes sign when you "negate" the contour (change its orientation) –  Alex R. Dec 27 '12 at 0:37
    
@Alex - thanks! you can make it an answer and I will accept it –  Belgi Dec 27 '12 at 0:39
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By definition, the line integral changes sign when you "negate" the contour (change its orientation)

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