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

@inproceedings{Bornemann2021AJF, title={A Jentzsch-Theorem for Kapteyn, Neumann, and General Dirichlet Series}, author={Folkmar A. Bornemann}, year={2021} }

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…

## References

SHOWING 1-10 OF 13 REFERENCES

Accelerated polynomial approximation of finite order entire functions by growth reduction

- MathematicsMath. Comput.
- 1997

The aim is to find sequences of functions which are the product of a polynomial of degree < n and an easy computable second factor and such that (f n ) n converges essentially faster to f on K than the sequence (P n * ) n of best approximating polynomials of degree ≤ n.

A Treatise on the Theory of Bessel Functions

- PhysicsNature
- 1945

THE memoir in which Bessel, the astronomer, examined in detail the functions which now bear his name was published in 1824, and was the outcome of his earlier researches concerning the expression of…

The Concrete Tetrahedron - Symbolic Sums, Recurrence Equations, Generating Functions, Asymptotic Estimates

- MathematicsTexts & Monographs in Symbolic Computation
- 2011

The book treats four mathematical concepts which play a fundamental role in many different areas of mathematics: symbolic sums, recurrence (difference) equations, generating functions, and asymptotic…

A Jentzsch-Type-Theorem

- Mathematics
- 2008

Suppose that $\matrix\sum^\infty_v=0\ a_vz^v$ is a power series with radius of convergence 1 and denote by $S_n(z)\matrix\sum^n_v=0\ a_vz^v$ its partial sums. In this paper, we investigate properties…

Theory of Functions of a Complex Variable

- MathematicsNature
- 1893

WHAT is the theory of functions about? This question may be heard now and again from a mathematical student; and if, by way of a pattial reply, it be said that the elements of the theory of functions…

The general theory of Dirichlet's series

- Mathematics
- 1915

V The Fundamental Properties of Analyt'ic Functions ; Taylor's, Latirent's, and Liouville's Theorems ; VI. The Theory of Residues ; Application to the Evaluation of Definite Integrals ; VIL. The…