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# 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)

Complex Argument, Contour

## References

Barnard, R. W.; Dayawansa, W.; Pearce, K.; and Weinberg, D. "Polynomials with Nonnegative Coefficients." Proc. Amer. Math. Soc. 113, 77-83, 1991.

## Referenced on Wolfram|Alpha

Variation of Argument

## Cite this as:

Weisstein, Eric W. "Variation of Argument." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/VariationofArgument.html