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A Continuous-Order Integral Operator for Maclaurin-type Reconstruction

Derek C. Braun

TL;DR

The paper develops a continuous-order generalization of the Maclaurin expansion by defining $\mathcal{T}[f](x)=\int_{0}^{\infty} \frac{D^{r}f(0)\,x^{r}}{\Gamma(r+1)}\,dr$ to reconstruct analytic functions from fractional-derivative data. It establishes an Euler–Maclaurin correction framework, introducing a correction series $\mathcal{E}[f](x)$ with explicit terms $\mathcal{E}_0,\mathcal{E}_1,\mathcal{E}_2$, to quantify and reduce the sum–integral discrepancy. Numerical experiments across a suite of analytic functions show that the uncorrected operator captures global structure with a modest offset, while incorporating the first three correction terms reduces the mean absolute error by about two orders of magnitude, to $\mathcal{O}(10^{-3})$. For monomials and Caputo-consistent boundary data, the method yields exact reconstructions, indicating a coherent framework for extending Maclaurin-type expansions to spectral representations in derivative order. The work points to a flexible, order-spectrum viewpoint in fractional calculus with potential implications for analytic reconstruction and spectral methods.

Abstract

I introduce a continuous-order integral analog of the Maclaurin expansion that reconstructs analytic functions from fractional derivative data. This operator replaces the discrete sum of integer-order derivatives in the classical Maclaurin expansion with an integral over fractional derivative orders, weighted by a Maclaurin-type kernel. I show that, under smoothness and decay assumptions, the discrepancy between this construction and the classical Maclaurin series is governed by the Euler--Maclaurin summation formula, and I define a corresponding correction series. Numerical experiments on a representative set of analytic functions show that the uncorrected operator reliably tracks the global structure of $f$, with a systematic, mostly constant offset and additional deviation near the origin. Adding the first three correction terms substantially reduces these discrepancies across the tested domains, lowering the mean absolute error by two orders of magnitude, from $10^{-1}$ to $10^{-3}$. For monomials, the order data collapse to a single atomic contribution; using Caputo-consistent boundary data, the operator reconstructs them exactly. Altogether, these results show that this continuous-order operator, together with its correction series, provides a coherent framework for extending the classical Maclaurin expansion to spectral representations in derivative order.

A Continuous-Order Integral Operator for Maclaurin-type Reconstruction

TL;DR

The paper develops a continuous-order generalization of the Maclaurin expansion by defining to reconstruct analytic functions from fractional-derivative data. It establishes an Euler–Maclaurin correction framework, introducing a correction series with explicit terms , to quantify and reduce the sum–integral discrepancy. Numerical experiments across a suite of analytic functions show that the uncorrected operator captures global structure with a modest offset, while incorporating the first three correction terms reduces the mean absolute error by about two orders of magnitude, to . For monomials and Caputo-consistent boundary data, the method yields exact reconstructions, indicating a coherent framework for extending Maclaurin-type expansions to spectral representations in derivative order. The work points to a flexible, order-spectrum viewpoint in fractional calculus with potential implications for analytic reconstruction and spectral methods.

Abstract

I introduce a continuous-order integral analog of the Maclaurin expansion that reconstructs analytic functions from fractional derivative data. This operator replaces the discrete sum of integer-order derivatives in the classical Maclaurin expansion with an integral over fractional derivative orders, weighted by a Maclaurin-type kernel. I show that, under smoothness and decay assumptions, the discrepancy between this construction and the classical Maclaurin series is governed by the Euler--Maclaurin summation formula, and I define a corresponding correction series. Numerical experiments on a representative set of analytic functions show that the uncorrected operator reliably tracks the global structure of , with a systematic, mostly constant offset and additional deviation near the origin. Adding the first three correction terms substantially reduces these discrepancies across the tested domains, lowering the mean absolute error by two orders of magnitude, from to . For monomials, the order data collapse to a single atomic contribution; using Caputo-consistent boundary data, the operator reconstructs them exactly. Altogether, these results show that this continuous-order operator, together with its correction series, provides a coherent framework for extending the classical Maclaurin expansion to spectral representations in derivative order.
Paper Structure (13 sections, 3 theorems, 37 equations, 6 figures)

This paper contains 13 sections, 3 theorems, 37 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The Continuous-Order Maclaurin-type Integral Operator
  3. Euler--Maclaurin Correction as a Theorem
  4. Methodology
  5. Results
  6. Exponential function: $f(x) = e^{x}$
  7. Rational function with a pole: $f(x) = \frac{1}{1-x}$
  8. Oscillatory entire function: $f(x) = \sin x$
  9. Quadratic exponential function: $f(x)=e^{x^{2}}$
  10. Rapidly decaying entire function: $f(x)=e^{-x^{2}}$
  11. Special function with oscillatory decay: $f(x) = J_{0}(x)$
  12. Monomials
  13. Conclusions and Discussion

Key Result

Lemma 2.3

For the kernel of the continuous-order operator eq:operator_def, assume that it is integrable on $[0,\infty)$ with the generalized Riemann (Henstock--Kurzweil) integral and that, for integers $n \ge 0$, the fractional derivative satisfies $D^{n}f(0)=f^{(n)}(0)$. The continuous-order operator then fo

Figures (6)

  • Figure 1: Reconstruction of $f(x) = e^x$ using $\mathcal{T}$ with added terms $\mathcal{E}_0$, $\mathcal{E}_1$ and $\mathcal{E}_2$. Top panels show the reconstruction and residuals, with mean absolute errors, over a representative interval; bottom panels zoom in on the origin.
  • Figure 2: Reconstruction of $f(x) = \tfrac{1}{1-x}$ using $\mathcal{T}$ with added terms $\mathcal{E}_0$, $\mathcal{E}_1$ and $\mathcal{E}_2$. Top panels show the reconstruction and residuals, with mean absolute errors, over a representative interval; bottom panels zoom in on the origin.
  • Figure 3: Reconstruction of $f(x) = \sin x$ using $\mathcal{T}$ with added terms $\mathcal{E}_0$, $\mathcal{E}_1$ and $\mathcal{E}_2$. Top panels show the reconstruction and residuals, with mean absolute errors, over a representative interval; bottom panels zoom in on the origin.
  • Figure 4: Reconstruction of the quadratic exponential function $f(x) = e^{x^2}$ using $\mathcal{T}$ with added terms $\mathcal{E}_0$, $\mathcal{E}_1$ and $\mathcal{E}_2$. An analytic continuation was used for fractional derivative data. Top panels show the reconstruction and residuals, with mean absolute errors, over a representative interval; bottom panels zoom in on the origin.
  • Figure 5: Reconstruction of the Gaussian function $f(x) = e^{-x^{2}}$ using $\mathcal{T}$ with added terms $\mathcal{E}_0$, $\mathcal{E}_1$ and $\mathcal{E}_2$. An analytic continuation was used for derivative data. Top panels show the reconstruction and residuals over a representative interval; bottom panels show the same near the origin. Mean absolute errors in the top right panel are for the interval.
  • ...and 1 more figures

Theorems & Definitions (10)

  • Definition 2.1
  • Remark 2.2: Choice of fractional derivative data
  • Lemma 2.3: The continuous-order operator formally recovers the classical Maclaurin series at discrete orders
  • Remark 2.4
  • Remark 3.1
  • Theorem 3.2: The Euler--Maclaurin summation formula applies to the continuous-order operator
  • proof
  • Definition 3.3: Correction series
  • Corollary 3.4: The continuous-order operator with correction reconstructs analytic functions exactly
  • proof