Variation of Argument

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Let [arg(f(z))] denote the change in the complex argument of a function f(z) around a contour gamma. Also let N denote the number of roots of f(z) in gamma and P denote the sum of the orders of all poles of f(z) lying inside gamma. Then

For example, the plots above shows the argument for a small circular contour gamma centered around z=0 for a function of the form f(z)=(z-1)/z^n (which has a single pole of order n and no roots in gamma) for n=1, 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 [arg(f(z))] in a given region R, break R into paths and find [arg(f(z))] for each path. On a circular arc

let f(z) be a polynomial P(z) of degree n. Then

Plugging in z=Re^(itheta) gives

So as R->infty,

and

For a real segment z=x,

For an imaginary segment z=iy,

See also

Complex Argument, Contour

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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 Resource. https://mathworld.wolfram.com/VariationofArgument.html

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