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Counting Zeros of Complex-Valued Harmonic Functions via Rouché's Theorem

Japheth Carlson

TL;DR

This work extends zero-counting techniques for complex-valued harmonic polynomials by applying a harmonic version of Rouché's theorem directly to non-circular critical curves. For the quadrinomial $f(z)=z^n+a z^k+b\overline{z}^k-1$ with $n>k$ and $a,b>0$, it derives conditions under which the total number of zeros is either $n$ or $n+2k$, and proves all zeros lie within two explicit annuli, with the inner annulus containing $k$ zeros. The analysis hinges on decomposing $f$ into analytic and co-analytic parts, studying the critical curve $|h'(z)|=|g'(z)|$, and using Rouché's theorem for harmonic functions to count zeros by region. The results generalize previous circular-geometry approaches and provide concrete localization that could inform further qualitative and quantitative studies of harmonic polynomials. The methodology promises applicability to broader harmonic families and potential extensions to poles and sharper geometric bounds.

Abstract

Rouché's Theorem is among the most useful results in complex analysis for counting zeros of analytic functions. Rouché's Theorem also admits a harmonic analogue for counting zeros of complex harmonic functions. Previously, this analogue has been applied primarily to closed curves of simple geometry, such as circles, to count zeros. We demonstrate that non-circular critical curves can serve as effective contours by applying a harmonic Rouché-type argument to determine the total number of zeros of the complex harmonic family given by $f(z) = z^n + az^k + b\overline{z}^k - 1 $, where $n>k\geq1$ and $a,b > 0$. Under explicit inequalities relating $a$ and $b$, we determine the total number of zeros is either $n$ or $n+2k$ (counted with multiplicity). We also prove the zeros of $f$ are confined to the union of two explicit annuli in the plane: an inner annulus containing $k$ zeros and an outer annulus containing the remainder.

Counting Zeros of Complex-Valued Harmonic Functions via Rouché's Theorem

TL;DR

This work extends zero-counting techniques for complex-valued harmonic polynomials by applying a harmonic version of Rouché's theorem directly to non-circular critical curves. For the quadrinomial with and , it derives conditions under which the total number of zeros is either or , and proves all zeros lie within two explicit annuli, with the inner annulus containing zeros. The analysis hinges on decomposing into analytic and co-analytic parts, studying the critical curve , and using Rouché's theorem for harmonic functions to count zeros by region. The results generalize previous circular-geometry approaches and provide concrete localization that could inform further qualitative and quantitative studies of harmonic polynomials. The methodology promises applicability to broader harmonic families and potential extensions to poles and sharper geometric bounds.

Abstract

Rouché's Theorem is among the most useful results in complex analysis for counting zeros of analytic functions. Rouché's Theorem also admits a harmonic analogue for counting zeros of complex harmonic functions. Previously, this analogue has been applied primarily to closed curves of simple geometry, such as circles, to count zeros. We demonstrate that non-circular critical curves can serve as effective contours by applying a harmonic Rouché-type argument to determine the total number of zeros of the complex harmonic family given by , where and . Under explicit inequalities relating and , we determine the total number of zeros is either or (counted with multiplicity). We also prove the zeros of are confined to the union of two explicit annuli in the plane: an inner annulus containing zeros and an outer annulus containing the remainder.

Paper Structure

This paper contains 9 sections, 9 theorems, 36 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Results
  4. Zero Counting
  5. Proof of Theorem \ref{['TheoremA']}
  6. Proof of Theorem \ref{['TheoremB']}
  7. Zero Locating
  8. Proof of Theorem \ref{['Annulus']}
  9. Conclusion

Key Result

Theorem 1.1

Let where $n$ and $k$ are natural numbers with $n>k$. Let $\epsilon > 0$. Suppose $0<a < (\frac{n-k}{n+k} - \epsilon)b$. Then, there exists $b_0$ such that, for all $b > b_0$, $f$ has $n+2k$ zeros.

Figures (5)

  • Figure 1: Zeros of $f_1$
  • Figure 2: Zeros of $f_2$
  • Figure 3: Critical Curve and Zeros of $f_1$
  • Figure 4: Critical Curve and Zeros of $f_2$
  • Figure 5: Radii binding location of zeros of $f(z) = z^9 + 7z^4 + 14\overline{z}^4 - 1$

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 9 more