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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$f(z) = z^n + az^k + b\overline{z}^k - 1 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mi>z</mi> <mi>n</mi> </msup> <mo>+</mo> <mi>a</mi> <msup> <mi>z</mi> <mi>k</mi> </msup> <mo>+</mo> <mi>b</mi> <msup> <mover> <mi>z</mi> <mo>¯</mo> </mover> <mi>k</mi> </msup> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$n>k\ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>></mo> <mi>k</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$a,b > 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. Under explicit inequalities relating a and b, we determine the total number of zeros is either n or <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$n+2k$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mi>k</mi> </mrow> </math></EquationSource> </InlineEquation> (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

期刊论文
发表日期：

2026年3月23日

查看全文

Japheth Carlson

摘要

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 $ $f (z) = z^{n} + a z^{k} + b {\bar{z}}^{k} - 1$ , where $n>k\ge 1$ $n > k \geq 1$ and $a,b > 0$ $a, b > 0$ . Under explicit inequalities relating a and b, we determine the total number of zeros is either n or $n+2k$ $n + 2 k$ (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.