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A Jentzsch-Theorem for Kapteyn, Neumann, and General Dirichlet Series

Folkmar Bornemann

Abstract

Comparing phase plots of truncated series solutions of Kepler's equation by Lagrange's power series with those by Bessel's Kapteyn series strongly suggest that a Jentzsch-type theorem holds true not only for the former but also for the latter series: each point of the boundary of the domain of convergence in the complex plane is a cluster point of zeros of sections of the series. We prove this result by studying properties of the growth function of a sequence of entire functions. For series, this growth function is computable in terms of the convergence abscissa of an associated general Dirichlet series. The proof then extends, besides including Jentzsch's classical result for power series, to general Dirichlet series, to Kapteyn, and to Neumann series of Bessel functions. Moreover, sections of Kapteyn and Neumann series generally exhibit zeros close to the real axis which can be explained, including their asymptotic linear density, by the theory of the distribution of zeros of entire functions.

A Jentzsch-Theorem for Kapteyn, Neumann, and General Dirichlet Series

Abstract

Comparing phase plots of truncated series solutions of Kepler's equation by Lagrange's power series with those by Bessel's Kapteyn series strongly suggest that a Jentzsch-type theorem holds true not only for the former but also for the latter series: each point of the boundary of the domain of convergence in the complex plane is a cluster point of zeros of sections of the series. We prove this result by studying properties of the growth function of a sequence of entire functions. For series, this growth function is computable in terms of the convergence abscissa of an associated general Dirichlet series. The proof then extends, besides including Jentzsch's classical result for power series, to general Dirichlet series, to Kapteyn, and to Neumann series of Bessel functions. Moreover, sections of Kapteyn and Neumann series generally exhibit zeros close to the real axis which can be explained, including their asymptotic linear density, by the theory of the distribution of zeros of entire functions.

Paper Structure

This paper contains 6 sections, 3 theorems, 52 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Growth functions and Jentzsch-type theorems
  3. A Jentzsch-type theorem for four types of series
  4. Zeros of sections of series of holonomic entire functions
  5. Some results from the theory of entire functions
  6. Application to general Dirichlet, Kapteyn and Neumann series

Key Result

theorem 1

Let $(f_n)$ be a sequence of entire functions with an admissible growth function $\mu$. If $(f_n)$ converges compactly to a non-constant holomorphic function $f$ on some open set $D$, then all points of the set are cluster points of zeros of the $f_n$.

Figures (4)

  • Figure 1: Phase plots for complex eccentricity $\epsilon$ of the series solution of Kepler's equation for $M=1/5$ truncated at $n=25$; left: power series \ref{['eq:lagrange']} with $\rho=0.84889\cdots$, right: Kapteyn series \ref{['eq:bessel']} with $\rho=1$; in both cases the boundary of the domain of convergence is shown as the solid black line, cf. Fig. \ref{['fig:Drho']} for the domains of convergence of Kapteyn series in general
  • Figure 2: Kapteyn series truncated at $n=25$; left: convergence level $\rho = 3/4$, right: $\rho=4/3$
  • Figure 3: Two series with convergence radius $\rho=5$ truncated at $n=25$; top: Neumann series, bottom: its associated power series, cf. MR0010746
  • Figure 4: Sublevel-sets $D_r$ of $\Omega(z)$, $r=\frac{1}{4},\frac{1}{2},\frac{3}{4},\ldots$

Theorems & Definitions (10)

  • definition 1
  • theorem 1
  • proof
  • definition 2
  • theorem 2
  • proof
  • theorem 3
  • proof
  • remark 1
  • remark 2