Argument Principle, Complex Analysis

Nov 2009
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I'm given that \(\displaystyle \frac{1}{2 \pi i} \int_{\partial D(0,3)} \frac{f'(z)}{f(z)}\,dz = 2\) and also that \(\displaystyle \frac{1}{2 \pi i} \int_{\partial D(0,3)} z \frac{f'(z)}{f(z)}\,dz = 2\). I know that \(\displaystyle f\) is analytic on the curve and on the interior region as well, and that \(\displaystyle f(z)\neq 0, \forall z \in \partial D(0,3)\).

The actual question of the problem aside, what does the second equation really tell me? I know that the first says that the winding number of \(\displaystyle f(D(0,3))\) about the origin is 2, and that because \(\displaystyle f\) is holomorphic in the disc there are no poles and so \(\displaystyle f\) must have two zeroes in \(\displaystyle D(0,3)\).

Eventually I'm trying to locate those zeroes, and there's a third equation as well that I imagine is supposed to help me along, but what more information can I glean from that second equation?

Thanks.
 

Laurent

MHF Hall of Honor
Aug 2008
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I'm given that \(\displaystyle \frac{1}{2 \pi i} \int_{\partial D(0,3)} \frac{f'(z)}{f(z)}\,dz = 2\) and also that \(\displaystyle \frac{1}{2 \pi i} \int_{\partial D(0,3)} z \frac{f'(z)}{f(z)}\,dz = 2\). I know that \(\displaystyle f\) is analytic on the curve and on the interior region as well, and that \(\displaystyle f(z)\neq 0, \forall z \in \partial D(0,3)\).

The actual question of the problem aside, what does the second equation really tell me?
If \(\displaystyle z_0\) is a zero of multiplicity \(\displaystyle k\), then \(\displaystyle f(z)\sim C(z-z_0)^k\) and \(\displaystyle f'(z)\sim Ck(z-z_0)^{k-1}\) as \(\displaystyle z\to z_0\), hence \(\displaystyle \frac{f'(z)}{f(z)}\sim\frac{k}{z-z_0}\) and \(\displaystyle \frac{z f'(z)}{f(z)}\sim\frac{kz_0}{z-z_0}\). Thus, the residue of \(\displaystyle z\mapsto\frac{f'(z)}{f(z)}\) at \(\displaystyle z_0\) is the multiplicity of \(\displaystyle z_0\) (as a zero), i.e. the "number of zeroes at \(\displaystyle z_0\)", while the residue of \(\displaystyle z\mapsto\frac{zf'(z)}{f(z)}\) is \(\displaystyle kz_0\), i.e. the "sum of zeroes at \(\displaystyle z_0\)" (counting multiplicity).

Thus, by the theorem of residues, the first integral is the number of zeroes inside the curve (counting multiplicity) and the second integral is the sum of these zeroes (counting multiplicity). With \(\displaystyle z^2\) you would get the sum of their squares, and so on.
 
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