A Laplace transform of irregular growth
Pith reviewed 2026-05-18 02:05 UTC · model grok-4.3
The pith
A Laplace transform can be constructed that does not have regular growth.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We give an example of a Laplace transform ∫_γ e^{ζ z} dμ(ζ) that does not have regular growth. This answers a question in Hayman's List.
What carries the argument
A chosen contour gamma and measure mu whose integral produces the Laplace transform and breaks the regular growth property.
If this is right
- Theorems that assume regular growth for all Laplace transforms of this type must add further restrictions to apply.
- The question listed in Hayman's List receives a negative answer.
- Irregular growth can arise even for integrals of this exponential form.
Where Pith is reading between the lines
- Similar constructions might be attempted for other integral representations in complex analysis to test growth properties.
- The example may influence how entire functions are classified when represented by contour integrals.
- One could test whether minor modifications to the contour restore regular growth.
Load-bearing premise
A contour gamma and measure mu exist such that the resulting Laplace transform integral fails to have regular growth.
What would settle it
Explicit calculation or estimation of the growth function for the constructed integral that shows violation of the regular growth definition would confirm the example; if the growth proves regular after all, the counterexample fails.
read the original abstract
We give an example of a Laplace transform $\int_\gamma e^{\zeta z} d\mu(\zeta)$ that does not have regular growth. This answers a question in Hayman's List
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript constructs a specific contour γ and measure μ such that the Laplace transform f(z) = ∫_γ e^{ζ z} dμ(ζ) fails to have regular growth, meaning its growth indicator h(θ) is discontinuous at some angle. This is presented as an explicit counterexample resolving a question from Hayman's list.
Significance. If the construction and estimates hold, the result supplies a concrete counterexample in the theory of Laplace transforms and entire functions, clarifying the boundary between regular and irregular growth. The paper's strength lies in its direct existence claim rather than a general theorem, but its impact depends on the rigor of the growth estimates.
major comments (1)
- [§4] §4 (Verification of irregular growth): The lower-bound estimate for log |f(re^{iθ})| along the direction of the claimed discontinuity must be shown to strictly exceed the contribution from the rest of the contour for large r; the current argument appears to provide only an upper bound consistent with irregularity rather than a matching lower bound that forces a jump in h(θ).
minor comments (2)
- [Introduction] The definition of 'regular growth' (presumably via the indicator h(θ)) should be recalled explicitly in the introduction with a reference to the standard definition in the literature.
- [Figure 1] Figure 1 (contour diagram) would benefit from labeling the angle θ_0 where the discontinuity occurs.
Simulated Author's Rebuttal
We thank the referee for the careful reading of the manuscript and the constructive comment on the growth estimates. We address the major comment below.
read point-by-point responses
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Referee: [§4] §4 (Verification of irregular growth): The lower-bound estimate for log |f(re^{iθ})| along the direction of the claimed discontinuity must be shown to strictly exceed the contribution from the rest of the contour for large r; the current argument appears to provide only an upper bound consistent with irregularity rather than a matching lower bound that forces a jump in h(θ).
Authors: We appreciate this observation. The argument in §4 isolates the contribution from a compact segment of γ supporting the measure μ, chosen so that the phase of e^{ζ z} is nearly constant and positive along the ray arg z = θ; this yields the lower bound log |f(re^{iθ})| ≥ (c - ε)r for any ε > 0 and all sufficiently large r. The integral over the complementary parts of γ is estimated by integration by parts or direct majorization, giving an upper bound O(r^α) with α < 1. The difference between these terms is therefore positive for large r and forces the discontinuity in h(θ). To make the domination explicit, we will add a short lemma comparing the two contributions uniformly in a small angular neighborhood of θ. revision: yes
Circularity Check
No circularity: explicit counterexample construction stands independently
full rationale
The paper supplies a concrete contour γ and measure μ as a direct counterexample to the regular-growth question from Hayman's List. No load-bearing step reduces to a self-definition, fitted input renamed as prediction, or self-citation chain; the existence claim rests on explicit estimates that are not forced by re-using the target property. The derivation is therefore self-contained against external benchmarks and receives the default non-circularity finding.
Axiom & Free-Parameter Ledger
Reference graph
Works this paper leans on
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[1]
M. A. Kaashoek & S. M. Verduyn Lunel, Completeness Theorems ans Characteristic Matrix Functions. Birkh\"auser, 2022. xvi+350 pp
work page 2022
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[2]
Hayman, Research problems in function theory
W.K. Hayman, Research problems in function theory. The Athlone Press [University of London], London, 1967. vii+56 pp
work page 1967
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[3]
W.K. Hayman & E. Lingham, Research Problems in Function Theory. Springer, 2019. viii+284 pp
work page 2019
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[4]
verbatim https://arxiv.org/pdf/1809.07200 verbatim
work page internal anchor Pith review Pith/arXiv arXiv
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[5]
B.Ja. Levin, Distribution of Zeros of Entire Functions, Revised edition, Vol 5 Translations of Mathematical Monographs, Amer. Math. Soc., 1972. xii+523 pp
work page 1972
discussion (0)
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