Variation of Argument

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.