Euler–Maclaurin Formula Overview

• The Euler–Maclaurin formula is a foundational result linking finite sums and integrals with explicit correction terms that account for boundary behavior and function smoothness.
• Its classical formulation uses Bernoulli polynomials to provide precise error estimates, while advanced versions extend these results to functions of bounded variation and multidimensional domains.
• Applications of the formula span numerical quadrature, series convergence acceleration, and spectral analysis, offering practical tools for both theoretical insights and computation.

The Euler–Maclaurin formula is a foundational result bridging discrete and continuous analysis, providing an explicit connection between finite sums and integrals through boundary corrections and remainder terms involving the smoothness of the underlying function. Originally developed for smooth functions, the formula admits rigorous generalizations to functions of bounded variation as well as multidimensional and singular settings, and it is of central importance in number theory, spectral geometry, asymptotic analysis, and numerical quadrature.

1. Fundamental Formulation and Classical Results

The classical first-order Euler–Maclaurin formula states that for any $f\in C^1([m,n])$ with $m<n$ integers,

$$S[f;m,n) := \sum_{k=m}^{n-1} f(k) = \int_m^n f(x)\,dx - \frac{1}{2}\bigl(f(n)-f(m)\bigr) + R[f],$$

where the remainder is

$$R[f] = \int_m^n f'(x)\,B_1(x-\lfloor x\rfloor)\,dx,$$

with $B_1(x) = x-\frac{1}{2}$ the first Bernoulli polynomial. The bound $|B_1|\le \frac{1}{2}$ on $[0,1]$ yields the explicit estimate

$|R[f]| \le \frac{1}{2} \int_m^n |f'(x)|\,dx.$

A symmetric variant is

$$\sum_{k=m}^{n-1}f(k) = \int_m^n f(x)\,dx + \frac{f(n)+f(m)}{2} + \tilde R[f], \quad \tilde R[f]=R[f]+\frac{f(n)-f(m)}{2}.$$

Higher-order versions incorporate derivatives of increasing order and even Bernoulli numbers, yielding a full asymptotic expansion for smooth enough $f$.

2. Functions of Bounded Variation and Rigorous Error Control

For $m<n$0 of bounded variation (BV) on $m<n$1, the first-order Euler–Maclaurin structure persists, but the control of the remainder is in terms of the total variation $m<n$2: $m<n$3 with

$m<n$4

For monotone $m<n$5, $m<n$6 so the estimate coincides with the classical monotone case.

A refined mid-value formula is, for the mid-value $m<n$7,

$m<n$8

where $m<n$9 is the 1-periodic extension of $$S[f;m,n) := \sum_{k=m}^{n-1} f(k) = \int_m^n f(x)\,dx - \frac{1}{2}\bigl(f(n)-f(m)\bigr) + R[f],$$0 vanishing at integers, and $$S[f;m,n) := \sum_{k=m}^{n-1} f(k) = \int_m^n f(x)\,dx - \frac{1}{2}\bigl(f(n)-f(m)\bigr) + R[f],$$1 is the Stieltjes measure induced by $$S[f;m,n) := \sum_{k=m}^{n-1} f(k) = \int_m^n f(x)\,dx - \frac{1}{2}\bigl(f(n)-f(m)\bigr) + R[f],$$2 (Marco et al., 2016). The explicit bound follows as $$S[f;m,n) := \sum_{k=m}^{n-1} f(k) = \int_m^n f(x)\,dx - \frac{1}{2}\bigl(f(n)-f(m)\bigr) + R[f],$$3 and $$S[f;m,n) := \sum_{k=m}^{n-1} f(k) = \int_m^n f(x)\,dx - \frac{1}{2}\bigl(f(n)-f(m)\bigr) + R[f],$$4.

These generalizations are elementary, require only Jordan decomposition and standard measure theory, and bypass Fourier analytic machinery entirely.

3. Multidimensional and Polyhedral Extensions

Euler–Maclaurin formulas generalize to sums over lattice points in polytopes and more general convex or polyhedral regions. For a convex polytope $$S[f;m,n) := \sum_{k=m}^{n-1} f(k) = \int_m^n f(x)\,dx - \frac{1}{2}\bigl(f(n)-f(m)\bigr) + R[f],$$5 with integer vertices and a smooth weight $$S[f;m,n) := \sum_{k=m}^{n-1} f(k) = \int_m^n f(x)\,dx - \frac{1}{2}\bigl(f(n)-f(m)\bigr) + R[f],$$6, weighted by interior, face, and vertex contributions (e.g., solid-angle weights), the sum over lattice points is

$$S[f;m,n) := \sum_{k=m}^{n-1} f(k) = \int_m^n f(x)\,dx - \frac{1}{2}\bigl(f(n)-f(m)\bigr) + R[f],$$7

where $$S[f;m,n) := \sum_{k=m}^{n-1} f(k) = \int_m^n f(x)\,dx - \frac{1}{2}\bigl(f(n)-f(m)\bigr) + R[f],$$8 is a normal-derivative operator of appropriate order acting on $$S[f;m,n) := \sum_{k=m}^{n-1} f(k) = \int_m^n f(x)\,dx - \frac{1}{2}\bigl(f(n)-f(m)\bigr) + R[f],$$9 (Brandolini et al., 2020, Brandolini et al., 2022, Floch et al., 2013).

In higher dimensions, infinite-order expansions involve multivariate Bernoulli polynomials and Lerch zeta functions. The general structure is that each $$R[f] = \int_m^n f'(x)\,B_1(x-\lfloor x\rfloor)\,dx,$$0-face of $$R[f] = \int_m^n f'(x)\,B_1(x-\lfloor x\rfloor)\,dx,$$1 contributes a term involving the $$R[f] = \int_m^n f'(x)\,B_1(x-\lfloor x\rfloor)\,dx,$$2th normal derivative of $$R[f] = \int_m^n f'(x)\,B_1(x-\lfloor x\rfloor)\,dx,$$3 on $$R[f] = \int_m^n f'(x)\,B_1(x-\lfloor x\rfloor)\,dx,$$4 integrated against a periodic Bernoulli polynomial associated to the lattice geometry (Brandolini et al., 2022). Explicit analytic machinery (e.g., Poisson summation and Fourier analysis) underpins the construction and error analysis in general polyhedral settings.

Modifications of the formula adapt to cones, wedges, and sectors, with combinatorial data (e.g., Todd polynomials, rational weights, etc.) entering via decomposition into simplicial cones (Guo et al., 2013, Jun et al., 2012).

4. Analytic, Algebraic, and Singular Generalizations

The Euler–Maclaurin formula admits several analytic generalizations:

• Singular and near-singular sums: For sums of functions with algebraic or logarithmic singularities at endpoints, or nearly singular kernels, the expansion splits into "singular" and "jump" components, with coefficients involving Hurwitz zeta values and digamma functions. For example, for $$R[f] = \int_m^n f'(x)\,B_1(x-\lfloor x\rfloor)\,dx,$$5 analytic near zero and small regularization parameter $$R[f] = \int_m^n f'(x)\,B_1(x-\lfloor x\rfloor)\,dx,$$6,

$$R[f] = \int_m^n f'(x)\,B_1(x-\lfloor x\rfloor)\,dx,$$7

reflects both singular and regularized “jump” phenomena (Wu, 2024).

• Algebraic Birkhoff factorization: The formula on lattice cones emerges via factorization in coalgebraic settings, with exponential sums decomposed into convolution quotients, and with Todd or Bernoulli–type coefficients structured by meromorphic germs (Guo et al., 2013).
• Multiple Bernoulli series: In higher dimensions, sums over lattices can be expanded into integrals involving multiple Bernoulli series, with distributional wall crossing and decomposition formulas, bridging number theory, and symplectic geometry (Boysal et al., 2010).
• Unified summation theory: All classical summation formulas, including Euler–Maclaurin, Poisson, Voronoi, and Taylor–Maclaurin, can be seen as expansions arising from a general “mother equation” via contour integration in the Mellin transform domain. Poisson summation and Euler–Maclaurin are thus exact/asymptotic duals (Jean-Christophe, 2016).

5. Computational and Applied Aspects

The Euler–Maclaurin formula is widely used for:

• Efficient approximation of sums: The formula yields high-accuracy estimates of finite and infinite sums, often with rigorous control of error provided by explicit remainder terms. For example, the first-order BV formula is robust even when $$R[f] = \int_m^n f'(x)\,B_1(x-\lfloor x\rfloor)\,dx,$$8 has jumps, and the explicit error is proportional to the total variation.
• Quadrature and numerical integration: Weighted Euler–Maclaurin expansions provide quadrature rules exact for polynomial data, with error bounds determined by the smoothness and variation of $$R[f] = \int_m^n f'(x)\,B_1(x-\lfloor x\rfloor)\,dx,$$9. In multidimensional settings, explicit area, edge, and vertex corrections enable high-fidelity lattice quadrature (Brandolini et al., 2020).
• Series convergence acceleration and asymptotics: Euler–Maclaurin correction terms accelerate the convergence of slowly convergent or even divergent series, and generalized expansions (e.g., alternative "integral-only" formulas as in (Pinelis, 2015)) can be computationally superior in large-precision computations.
• Theoretical physics and spectral theory: In quantum statistical systems, the formula underlies analytic approximations of partition functions and spectral densities, including for polygonal domains and in finite-temperature calculations (Guo et al., 2020).

6. Extensions to Geometry, Representation Theory, and Stochastic Processes

The formula has deep connections and extended versions in several domains:

• Equivariant toric geometry: In the study of lattice polytopes and toric varieties, the Euler–Maclaurin formula generalizes to yield closed-form expressions for lattice-point counts and various cohomological and K-theoretic invariants, with connections to Hirzebruch–Riemann–Roch and motivic characteristic classes (Cappell et al., 2024).
• Spectral theory and semiclassical analysis: Asymptotic Euler–Maclaurin formulas provide full expansions for trace formulas of Toeplitz and other operators whose joint spectrum lies on lattice polytopes, with all correction terms encapsulated via normal derivatives localized to lower-dimensional faces (Floch et al., 2013).
• Stochastic processes and Riemann–Stieltjes integration: Recent generalizations recast the formula in the language of signatures of rectifiable paths (including "flip-" and "sawtooth-signatures"), allowing the transition from discrete sums to stochastic pathwise integrals with optimal remainder control; Bernoulli numbers emerge as solutions to repeated integration by parts minimizing error at each level (Bellingeri et al., 2024).

7. Concluding Remarks and Context

The scope of the Euler–Maclaurin formula, from its classical one-dimensional presentation to its modern extensions in higher dimensions, nonsmooth functions, and singular situations, is both broad and unified under the theme of connecting discrete summation and continuous integration with explicit correction terms. Its variants allow precise analysis and numerical computation in analytic number theory, spectral problems, representation theory, combinatorics, symplectic geometry, stochastic calculus, and statistical physics. The key feature in all settings is the capture and quantification of the arithmetic and analytic errors arising in the passage between sums and integrals, with rigorous error bounds dictated by smoothness, variation, or singularity structure (Marco et al., 2016, Pinelis, 2015, Buchheit et al., 2021, Floch et al., 2013, Jean-Christophe, 2016).