Borwein Integral: Threshold Phenomenon

• Borwein integrals are a family of definite integrals defined using products of sinc functions, exhibiting a constant value (π or π/2) for initial cases until a threshold is reached.
• The analysis employs Fourier techniques and residue theory to explain how a precise combinatorial sign pattern leads to abrupt deviations from the constant behavior.
• Extensions of the theory generalize to various frequency sets and novel dominance patterns, yielding explicit correction terms and computational insights.

Borwein integrals are a family of definite integrals built from products of the cardinal sine function,

$$\operatorname{sinc}(x)=\frac{\sin x}{x}, \qquad \operatorname{sinc}(0)=1,$$

discovered by David Borwein and Jonathan Borwein. In the classical odd-denominator family,

$$I_n=\int_{-\infty}^{\infty}\prod_{j=1}^n \operatorname{sinc}\!\left(\frac{x}{2j-1}\right)\,dx,$$

the first several values are exactly $\pi$, and only later does the pattern fail. This “constant for a long time, then suddenly smaller” behavior has become the defining feature of the subject. Subsequent work has supplied Fourier-analytic, residue-theoretic, and computational explanations, and has extended the phenomenon to more general frequency sets and to regimes in which the deviation from the constant value is explicitly computable (Labora et al., 2024, Bäsel et al., 2015).

1. Classical family and normalizations

The classical Borwein integrals considered in the recent residue-theoretic treatment are

$$I_n=\int_{-\infty}^{\infty}\prod_{j=1}^n \operatorname{sinc}\!\left(\frac{x}{2j-1}\right)\,dx.$$

The first values displayed there are

$$I_1=\int_{-\infty}^{\infty}\operatorname{sinc}(x)\,dx=\pi,$$

$$I_2=\int_{-\infty}^{\infty}\operatorname{sinc}(x)\operatorname{sinc}(x/3)\,dx=\pi,$$

$$I_3=\int_{-\infty}^{\infty}\operatorname{sinc}(x)\operatorname{sinc}(x/3)\operatorname{sinc}(x/5)\,dx=\pi,$$

and

$$I_4=\int_{-\infty}^{\infty}\operatorname{sinc}(x)\operatorname{sinc}(x/3)\operatorname{sinc}(x/5)\operatorname{sinc}(x/7)\,dx=\pi.$$

More generally, the identity holds for $n\le 7$, while for $n\ge 8$ the integral is strictly less than $$I_n=\int_{-\infty}^{\infty}\prod_{j=1}^n \operatorname{sinc}\!\left(\frac{x}{2j-1}\right)\,dx,$$0 for the standard odd-denominator family (Labora et al., 2024).

A parallel normalization, used in computational work on the same phenomenon, is based on

$$I_n=\int_{-\infty}^{\infty}\prod_{j=1}^n \operatorname{sinc}\!\left(\frac{x}{2j-1}\right)\,dx,$$1

together with

$$I_n=\int_{-\infty}^{\infty}\prod_{j=1}^n \operatorname{sinc}\!\left(\frac{x}{2j-1}\right)\,dx,$$2

In that convention the constant regime is $$I_n=\int_{-\infty}^{\infty}\prod_{j=1}^n \operatorname{sinc}\!\left(\frac{x}{2j-1}\right)\,dx,$$3, not $$I_n=\int_{-\infty}^{\infty}\prod_{j=1}^n \operatorname{sinc}\!\left(\frac{x}{2j-1}\right)\,dx,$$4. The most studied specialization in that framework takes

$$I_n=\int_{-\infty}^{\infty}\prod_{j=1}^n \operatorname{sinc}\!\left(\frac{x}{2j-1}\right)\,dx,$$5

so the integrand is

$$I_n=\int_{-\infty}^{\infty}\prod_{j=1}^n \operatorname{sinc}\!\left(\frac{x}{2j-1}\right)\,dx,$$6

(Bäsel et al., 2015).

2. The threshold phenomenon

The abrupt transition at $$I_n=\int_{-\infty}^{\infty}\prod_{j=1}^n \operatorname{sinc}\!\left(\frac{x}{2j-1}\right)\,dx,$$7 admits a precise combinatorial description. In the residue-theoretic formulation one writes

$$I_n=\int_{-\infty}^{\infty}\prod_{j=1}^n \operatorname{sinc}\!\left(\frac{x}{2j-1}\right)\,dx,$$8

For $$I_n=\int_{-\infty}^{\infty}\prod_{j=1}^n \operatorname{sinc}\!\left(\frac{x}{2j-1}\right)\,dx,$$9, the sign of $\pi$0 is determined entirely by $\pi$1. The reason is the bound

$\pi$2

so a positive first sign cannot be overturned by the remaining terms. In that regime the residue sum collapses in such a way that

$\pi$3

(Labora et al., 2024).

The first obstruction appears at $\pi$4. The paper isolates the specific combination

$\pi$5

which is negative even though the first sign is positive. This is exactly the failure mechanism: sign patterns that would have contributed to the constant regime are now “misclassified.” The resulting correction is explicit: $\pi$6 and the residue-theoretic paper also gives the rational correction term in fully expanded form. The same sign-pattern mechanism governs all later departures from the constant value (Labora et al., 2024).

3. Fourier support and residue theory

The classical explanation of the Borwein phenomenon uses Fourier Analysis techniques. In the summary given in the residue-theoretic paper, the standard picture is that $\pi$7-functions have compactly supported Fourier transforms, products of sinc factors correspond to convolutions, and the equality with $\pi$8 persists while the relevant support constraints remain in the stable regime. Once those constraints fail, the constant value breaks (Labora et al., 2024).

The residue-theoretic reformulation replaces Fourier support by contour closure and residue extraction. Writing

$\pi$9

the paper decomposes $$I_n=\int_{-\infty}^{\infty}\prod_{j=1}^n \operatorname{sinc}\!\left(\frac{x}{2j-1}\right)\,dx.$$0 as

$$I_n=\int_{-\infty}^{\infty}\prod_{j=1}^n \operatorname{sinc}\!\left(\frac{x}{2j-1}\right)\,dx.$$1

where $$I_n=\int_{-\infty}^{\infty}\prod_{j=1}^n \operatorname{sinc}\!\left(\frac{x}{2j-1}\right)\,dx.$$2 decays on a large upper semicircle and $$I_n=\int_{-\infty}^{\infty}\prod_{j=1}^n \operatorname{sinc}\!\left(\frac{x}{2j-1}\right)\,dx.$$3 decays on a large lower semicircle. With the contours

$$I_n=\int_{-\infty}^{\infty}\prod_{j=1}^n \operatorname{sinc}\!\left(\frac{x}{2j-1}\right)\,dx.$$4

$$I_n=\int_{-\infty}^{\infty}\prod_{j=1}^n \operatorname{sinc}\!\left(\frac{x}{2j-1}\right)\,dx.$$5

the residue theorem yields

$$I_n=\int_{-\infty}^{\infty}\prod_{j=1}^n \operatorname{sinc}\!\left(\frac{x}{2j-1}\right)\,dx.$$6

After expanding

$$I_n=\int_{-\infty}^{\infty}\prod_{j=1}^n \operatorname{sinc}\!\left(\frac{x}{2j-1}\right)\,dx.$$7

the product becomes a sum of $$I_n=\int_{-\infty}^{\infty}\prod_{j=1}^n \operatorname{sinc}\!\left(\frac{x}{2j-1}\right)\,dx.$$8 terms of the form

$$I_n=\int_{-\infty}^{\infty}\prod_{j=1}^n \operatorname{sinc}\!\left(\frac{x}{2j-1}\right)\,dx.$$9

with $$I_1=\int_{-\infty}^{\infty}\operatorname{sinc}(x)\,dx=\pi,$$0. The function $$I_1=\int_{-\infty}^{\infty}\operatorname{sinc}(x)\,dx=\pi,$$1 is assembled from the terms with $$I_1=\int_{-\infty}^{\infty}\operatorname{sinc}(x)\,dx=\pi,$$2, and $$I_1=\int_{-\infty}^{\infty}\operatorname{sinc}(x)\,dx=\pi,$$3 from those with $$I_1=\int_{-\infty}^{\infty}\operatorname{sinc}(x)\,dx=\pi,$$4. For a typical summand,

$$I_1=\int_{-\infty}^{\infty}\operatorname{sinc}(x)\,dx=\pi,$$5

This converts the Borwein integral into an explicit finite sign-sum over the positive $$I_1=\int_{-\infty}^{\infty}\operatorname{sinc}(x)\,dx=\pi,$$6, making the threshold phenomenon a problem in residue calculus and sign combinatorics rather than only a problem in Fourier support (Labora et al., 2024).

4. Exact corrections and the “tiny numbers” regime

A theorem of Borwein and Jon Borwein, used in the computational study of sinc integrals, gives an exact threshold criterion. If $$I_1=\int_{-\infty}^{\infty}\operatorname{sinc}(x)\,dx=\pi,$$7 and

$$I_1=\int_{-\infty}^{\infty}\operatorname{sinc}(x)\,dx=\pi,$$8

then

$$I_1=\int_{-\infty}^{\infty}\operatorname{sinc}(x)\,dx=\pi,$$9

equivalently

$$I_2=\int_{-\infty}^{\infty}\operatorname{sinc}(x)\operatorname{sinc}(x/3)\,dx=\pi,$$0

Thus the normalized integral stays exactly constant as long as the leading frequency dominates the cumulative tail (Bäsel et al., 2015).

When the sum first exceeds the threshold, the drop is explicit. Under the size condition

$$I_2=\int_{-\infty}^{\infty}\operatorname{sinc}(x)\operatorname{sinc}(x/3)\,dx=\pi,$$1

together with

$$I_2=\int_{-\infty}^{\infty}\operatorname{sinc}(x)\operatorname{sinc}(x/3)\,dx=\pi,$$2

the exact correction is

$$I_2=\int_{-\infty}^{\infty}\operatorname{sinc}(x)\operatorname{sinc}(x/3)\,dx=\pi,$$3

hence

$$I_2=\int_{-\infty}^{\infty}\operatorname{sinc}(x)\operatorname{sinc}(x/3)\,dx=\pi,$$4

where

$$I_2=\int_{-\infty}^{\infty}\operatorname{sinc}(x)\operatorname{sinc}(x/3)\,dx=\pi,$$5

In the odd-reciprocal specialization,

$$I_2=\int_{-\infty}^{\infty}\operatorname{sinc}(x)\operatorname{sinc}(x/3)\,dx=\pi,$$6

and

$$I_2=\int_{-\infty}^{\infty}\operatorname{sinc}(x)\operatorname{sinc}(x/3)\,dx=\pi,$$7

The striking feature is that $$I_2=\int_{-\infty}^{\infty}\operatorname{sinc}(x)\operatorname{sinc}(x/3)\,dx=\pi,$$8 can be extraordinarily small. The paper’s headline example takes $$I_2=\int_{-\infty}^{\infty}\operatorname{sinc}(x)\operatorname{sinc}(x/3)\,dx=\pi,$$9, for which

$$I_3=\int_{-\infty}^{\infty}\operatorname{sinc}(x)\operatorname{sinc}(x/3)\operatorname{sinc}(x/5)\,dx=\pi,$$0

so the corresponding normalized integral involves

$$I_3=\int_{-\infty}^{\infty}\operatorname{sinc}(x)\operatorname{sinc}(x/3)\operatorname{sinc}(x/5)\,dx=\pi,$$1

sinc factors and satisfies

$$I_3=\int_{-\infty}^{\infty}\operatorname{sinc}(x)\operatorname{sinc}(x/3)\operatorname{sinc}(x/5)\,dx=\pi,$$2

with

$$I_3=\int_{-\infty}^{\infty}\operatorname{sinc}(x)\operatorname{sinc}(x/3)\operatorname{sinc}(x/5)\,dx=\pi,$$3

The same paper computes these values by Euler–Maclaurin summation, both for the threshold index $$I_3=\int_{-\infty}^{\infty}\operatorname{sinc}(x)\operatorname{sinc}(x/3)\operatorname{sinc}(x/5)\,dx=\pi,$$4 and for the logarithm of the factorial ratio entering $$I_3=\int_{-\infty}^{\infty}\operatorname{sinc}(x)\operatorname{sinc}(x/3)\operatorname{sinc}(x/5)\,dx=\pi,$$5 (Bäsel et al., 2015).

5. Generalizations beyond the odd reciprocals

The residue-theoretic analysis yields a clean extension from the classical odd-denominator sequence to arbitrary decreasing frequencies. Let $$I_3=\int_{-\infty}^{\infty}\operatorname{sinc}(x)\operatorname{sinc}(x/3)\operatorname{sinc}(x/5)\,dx=\pi,$$6 be a non-increasing sequence of positive real numbers such that for some $$I_3=\int_{-\infty}^{\infty}\operatorname{sinc}(x)\operatorname{sinc}(x/3)\operatorname{sinc}(x/5)\,dx=\pi,$$7,

$$I_3=\int_{-\infty}^{\infty}\operatorname{sinc}(x)\operatorname{sinc}(x/3)\operatorname{sinc}(x/5)\,dx=\pi,$$8

Then for every $$I_3=\int_{-\infty}^{\infty}\operatorname{sinc}(x)\operatorname{sinc}(x/3)\operatorname{sinc}(x/5)\,dx=\pi,$$9,

$$I_4=\int_{-\infty}^{\infty}\operatorname{sinc}(x)\operatorname{sinc}(x/3)\operatorname{sinc}(x/5)\operatorname{sinc}(x/7)\,dx=\pi.$$0

but

$$I_4=\int_{-\infty}^{\infty}\operatorname{sinc}(x)\operatorname{sinc}(x/3)\operatorname{sinc}(x/5)\operatorname{sinc}(x/7)\,dx=\pi.$$1

This theorem isolates the one-dominant-frequency mechanism underlying the classical Borwein plateau (Labora et al., 2024).

The same paper also gives a new “three dominant frequencies” generalization. If $$I_4=\int_{-\infty}^{\infty}\operatorname{sinc}(x)\operatorname{sinc}(x/3)\operatorname{sinc}(x/5)\operatorname{sinc}(x/7)\,dx=\pi.$$2 and

$$I_4=\int_{-\infty}^{\infty}\operatorname{sinc}(x)\operatorname{sinc}(x/3)\operatorname{sinc}(x/5)\operatorname{sinc}(x/7)\,dx=\pi.$$3

satisfy

$$I_4=\int_{-\infty}^{\infty}\operatorname{sinc}(x)\operatorname{sinc}(x/3)\operatorname{sinc}(x/5)\operatorname{sinc}(x/7)\,dx=\pi.$$4

then

$$I_4=\int_{-\infty}^{\infty}\operatorname{sinc}(x)\operatorname{sinc}(x/3)\operatorname{sinc}(x/5)\operatorname{sinc}(x/7)\,dx=\pi.$$5

Two examples are highlighted. If $$I_4=\int_{-\infty}^{\infty}\operatorname{sinc}(x)\operatorname{sinc}(x/3)\operatorname{sinc}(x/5)\operatorname{sinc}(x/7)\,dx=\pi.$$6 and $$I_4=\int_{-\infty}^{\infty}\operatorname{sinc}(x)\operatorname{sinc}(x/3)\operatorname{sinc}(x/5)\operatorname{sinc}(x/7)\,dx=\pi.$$7, then

$$I_4=\int_{-\infty}^{\infty}\operatorname{sinc}(x)\operatorname{sinc}(x/3)\operatorname{sinc}(x/5)\operatorname{sinc}(x/7)\,dx=\pi.$$8

For the factorial sequence $$I_4=\int_{-\infty}^{\infty}\operatorname{sinc}(x)\operatorname{sinc}(x/3)\operatorname{sinc}(x/5)\operatorname{sinc}(x/7)\,dx=\pi.$$9, dominated convergence gives

$n\le 7$0

These formulas show that the Borwein mechanism is not restricted to the odd reciprocals and, in the residue framework, extends naturally to more elaborate dominance patterns (Labora et al., 2024).

6. Nomenclature and distinct Borwein-related integrals

The expression “Borwein integral” is not completely unique in the literature. In the analytic proof of the Borwein Conjecture, the term is used for a contour coefficient integral arising from Cauchy’s formula: $n\le 7$1 or, on $n\le 7$2,

$n\le 7$3

That paper states that the resulting coefficient integral and its analysis are referred to as the Borwein Integral in the context of the conjecture. This usage is analytically unrelated to the classical sinc-product family, although it belongs to the same Borwein corpus (Wang, 2019).

A second distinct object is the Borwein–Broadhurst dilogarithmic integral

$n\le 7$4

which was conjectured by Borwein and Broadhurst to equal $n\le 7$5 and was later proved to satisfy

$n\le 7$6

In that setting the integral is tied to Clausen functions, Dedekind zeta values, and the volume of an ideal tetrahedron in hyperbolic space $n\le 7$7 (Cvijović, 2010).

This suggests that local context is essential. In contemporary analysis, “Borwein integrals” in the plural usually denotes the sinc-product family, but singular usages can denote a coefficient-extraction contour integral or another Borwein-related definite integral. The unifying feature is not a single formula, but a characteristic Borwein style: experimentally striking identities, sharp threshold behavior, and exact evaluation by analytic structure rather than by elementary antiderivatives.