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Date: 26-12-2018
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Date: 18-11-2018
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Date: 18-11-2018
1180
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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) |
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) |
let be a polynomial of degree . Then
(3) |
|||
(4) |
Plugging in gives
(5) |
So as ,
(6) |
(7) |
and
(8) |
For a real segment ,
(9) |
For an imaginary segment ,
(10) |
REFERENCES:
Barnard, R. W.; Dayawansa, W.; Pearce, K.; and Weinberg, D. "Polynomials with Nonnegative Coefficients." Proc. Amer. Math. Soc. 113, 77-83, 1991.
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مخاطر خفية لمكون شائع في مشروبات الطاقة والمكملات الغذائية
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"آبل" تشغّل نظامها الجديد للذكاء الاصطناعي على أجهزتها
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المجمع العلميّ يُواصل عقد جلسات تعليميّة في فنون الإقراء لطلبة العلوم الدينيّة في النجف الأشرف
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