Let denote the change in the complex argument of a function around a contour . Also let denote the number of roots of in and denote the sum of the orders of all poles of lying inside . Then
(1)
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For example, the plots above shows the argument for a small circular contour centered around for a function of the form (which has a single pole of order and no roots in ) for , 2, and 3.
Note that the complex argument must change continuously, so any "jumps" that occur as the contour crosses branch cuts must be taken into account.
To find in a given region , break into paths and find for each path. On a circular arc
(2)
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let be a polynomial of degree . Then
(3)
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(4)
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Plugging in gives
(5)
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So as ,
(6)
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(7)
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and
(8)
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For a real segment ,
(9)
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For an imaginary segment ,
(10)
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