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Convergence criteria for Frullani-type integrals involving differences of cosines

This paper provides a complete classification of convergence conditions and derives explicit closed-form evaluations for a family of Frullani-type improper integrals involving powers of cosine and sine differences, while also uncovering associated combinatorial identities.

Original authors: Atiratch Laoharenoo, Chanatip Sujsuntinukul

Published 2026-05-27

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CC BY 4.0

Original authors: Atiratch Laoharenoo, Chanatip Sujsuntinukul

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are standing at the edge of a vast, infinite river. This river represents a mathematical problem called an integral, which is essentially a way of measuring the total "area" or "amount" of something flowing over an infinite distance.

The specific river this paper studies is made of waves (specifically, cosine waves, which look like smooth hills and valleys). The authors are looking at a very specific type of water flow: the difference between two waves of different sizes, raised to a power, and divided by a number that gets smaller and smaller as you go further out.

Here is the breakdown of their journey, explained simply:

1. The Setup: The "Frullani" River

Long ago, a mathematician named Frullani discovered a neat trick. If you take two waves, stretch one out and squeeze the other, and look at the difference between them, the total area under the curve often relates to a simple logarithm (a number that grows slowly, like the number of digits in a large number).

The authors of this paper asked: "What happens if we make the waves more complicated?"

Instead of just one wave minus another, what if we take the difference and cube it (or raise it to any power $p$ )?
And what if we divide by a number that changes at different speeds (power $q$ )?

They wanted to know two things:

Does the river have a finite amount of water? (Does the integral converge?)
If it does, exactly how much water is there? (Can we find a closed-form answer?)

2. The Two Big Hurdles

The authors found that this river is tricky because it has two dangerous spots where the water behaves wildly:

The Source ( $x=0$ ): Right at the start, the waves might crash into each other in a way that creates a "singularity" (a point where the math blows up).
The Horizon ( $x=\infty$ ): As you look infinitely far away, the waves keep oscillating forever. Sometimes they cancel each other out perfectly; other times, they add up to an infinite amount of water.

3. The Map: When Does the River Flow?

The authors spent a lot of time drawing a map to tell you exactly when the river is safe to cross (converges) and when it is a flood (diverges).

The "Even vs. Odd" Rule:
- If you raise the wave difference to an even power (like squaring it), the result is always positive (like a hill). If the denominator isn't strong enough to pull the water down, the river flows to infinity.
- If you raise it to an odd power, the waves can be positive or negative. This allows them to cancel each other out, like a tide going in and out. This cancellation is the key to making the river finite.
The "Secret Ratio" (The $F_p$ Set):
For the odd-power cases, the authors discovered a hidden pattern. Whether the river flows or floods depends on the ratio of the two wave speeds ( $\alpha$ and $\beta$ ).
They found a special list of "forbidden ratios" (which they called the set $F_p$ ). If your wave speeds match these ratios, the waves cancel out in a way that leaves a constant "drift," causing the river to flood. If your ratio is not on this list, the waves cancel out perfectly, and the river is finite.

4. The Treasure: Finding the Exact Amount

Once they knew the river was safe (convergent), they wanted to know the exact amount of water.

They used a mathematical "sledgehammer" called Abel's summation. Imagine taking a complex, messy pile of waves and breaking them down into a neat stack of simple, single waves.
Once broken down, they could apply old, known rules to calculate the total area.
The Result: They produced a formula that gives the exact answer for any safe combination of wave speeds and powers. It's like having a universal calculator for these specific wave problems.

5. A Special Side Quest: The "Zero" Wave

They also looked at a special case where one of the waves is completely flat (speed = 0).

This turned out to be a goldmine for combinatorics (the math of counting and arranging things).
The numbers they found in the wave formulas matched up with patterns used in counting games and probability. It's as if the physics of the waves secretly whispered a secret code about how to count objects.

6. The Sine Twins

Finally, they asked: "What if we used Sine waves instead of Cosine waves?"

Sine waves start at zero, while Cosine waves start at a peak. This small difference changes the rules.
They found that for Sine waves, the "Even vs. Odd" rule flips. The river only flows safely if the power is odd. If it's even, the river floods.
They provided a sketch of the map for these Sine rivers, showing that the logic is similar but the details are slightly different.

Summary

In short, this paper is a comprehensive guidebook for a specific type of infinite wave integral.

The Problem: Do these complex wave differences add up to a finite number?
The Discovery: Yes, but only under very specific conditions involving whether the power is even or odd, and the specific ratio of the wave speeds.
The Solution: They provided a complete list of "safe" conditions and a formula to calculate the exact answer whenever it is safe.

They didn't just guess; they built a rigorous mathematical bridge that connects the behavior of these waves to known rules, revealing hidden patterns in the numbers along the way.

Technical Summary: Convergence Criteria for Frullani-type Integrals Involving Differences of Cosines

Problem Statement
This paper investigates the family of improper integrals defined by:
$I_{p,q}(\alpha, \beta) := \int_0^\infty \frac{(\cos \alpha x - \cos \beta x)^p}{x^q} \, dx$
where $p, q \in \mathbb{N}$ and $\alpha, \beta \in \mathbb{R}$ with $|\alpha| \neq |\beta|$ . While the classical Frullani integral ( $p=1, q=1$ ) is well-understood, and specific cases (such as $p \ge q$ with even $q$ ) were previously analyzed by Laoharenoo and Boonklurb (2022), the general convergence behavior and explicit evaluation for all parameter ranges remained unclassified. The integrand presents dual challenges: a potential singularity at $x=0$ and the infinite upper limit, complicated by the oscillatory nature of the cosine terms raised to arbitrary powers.

Methodology
The authors employ a combination of asymptotic analysis, trigonometric power expansions, and summation techniques to address the problem:

Trigonometric Expansion: The core of the analysis involves expanding $(\cos \alpha x - \cos \beta x)^p$ $(cos α x - cos β x)^{p}$ into a linear combination of unpowered cosine terms (or sines for the sine-difference variant). The paper utilizes two distinct expansion strategies:
- One expansion (Lemma 3.6) is optimized for determining convergence by identifying constant terms and frequency cancellations.
- A second expansion (Lemma 4.1) is designed for explicit evaluation, utilizing binomial coefficients and specific combinatorial indices.
Singularity Analysis:
- For $x \to 0$ , Taylor expansions are used to determine the order of the zero of the numerator. This establishes that the integrand behaves like $x^{2p-q}$ near the origin.
- For $x \to \infty$ , the authors analyze the decay rate relative to $x^{-q}$ and the oscillatory cancellation properties.
Abel's Summation: To evaluate the integrals, the authors apply Abel's summation formula to the series of cosine terms. This technique transforms the integral of a sum of cosines into a sum of logarithmic terms (for $q=1$ ) or utilizes the known values of $\int_0^\infty \frac{\sin kx}{x} dx$ (for even $q$ ).
Combinatorial Identities: In the special case where $\alpha = 0$ , the coefficients of the expansion are analyzed to derive new combinatorial identities involving binomial coefficients and Vandermonde-like determinants.

Key Contributions and Results

1. Complete Convergence Classification
The paper provides a rigorous classification of convergence based on the parity of $p$ and the relationship between $p$ and $q$ :

Case $q = 1$ :
- If $p$ is even, the integral diverges for all $|\alpha| \neq |\beta|$ . This is proven by showing the expansion contains a non-zero constant term, leading to logarithmic divergence.
- If $p$ is odd, the integral diverges if and only if the ratio $|\alpha/\beta|$ (or its reciprocal) belongs to a specific set $F_p$ . This set $F_p$ is constructed from rational numbers derived from the indices of the trigonometric expansion where frequency cancellation produces a constant term. If the ratio is not in $F_p$ , the integral converges.
Case $q \ge 2$ :
- The integral diverges if $q \ge 2p + 1$ . The singularity at $x=0$ is too strong for the numerator to compensate.
- The integral converges if $2 \le q \le 2p$ . In this range, the behavior at $x=0$ is integrable, and the decay at infinity is sufficient.

2. Explicit Closed-Form Evaluations
For all convergent cases, the authors derive explicit formulas:

For $q=1$ (odd $p$ ): The value is expressed as a finite sum of logarithmic terms involving the frequencies of the expansion components, weighted by partial sums of the expansion coefficients.
For $q \ge 2$ :
- If $q$ is even, the value is a finite sum involving powers of the frequencies and binomial coefficients, scaled by $\pi$ .
- If $q$ is odd, the value is a finite sum of logarithmic terms, similar to the $q=1$ case but derived via differentiation of the integrand.

3. Special Case $\alpha = 0$ and Combinatorial Identities
When $\alpha = 0$ , the integral simplifies to involving $(1 - \cos \beta x)^p$ . The authors derive explicit forms for the coefficients of the cosine expansion. By equating these coefficients with those derived from the half-angle formula and Euler's identity, the paper establishes new combinatorial identities relating sums of binomial coefficients to specific rational numbers.

4. Extension to Sine Differences
The methodology is extended to the analogous family $K_{p,q}(\alpha, \beta)$ involving $(\sin \alpha x - \sin \beta x)^p$ .

For $q=1$ , convergence occurs if and only if $p$ is odd.
For $q \ge 2$ , convergence occurs if and only if $q \le p$ .
The paper notes that the convergence criteria for sine differences differ slightly from those for cosine differences, particularly regarding the parity of $p$ .

Significance and Claims
The authors state that this work provides the first systematic analysis of the general family $I_{p,q}(\alpha, \beta)$ , filling a gap in the literature where such integrals were previously unrecorded in major compendiums. The paper claims to:

Establish a complete classification of convergence parameters, resolving cases left open by previous studies (specifically the odd $p$ cases for $q=1$ and the general $q \ge 2$ cases).
Derive explicit closed-form evaluations for all convergent scenarios, extending the work of Laoharenoo and Boonklurb (2022).
Reveal a natural connection between these analytic integrals and combinatorial identities, particularly in the $\alpha=0$ case.
Demonstrate that the convergence behavior is highly sensitive to the parity of $p$ and the specific rational relationships between $\alpha$ and $\beta$ in the $q=1$ case.

The paper concludes by noting that while the relationship between the cosine and sine families is not direct due to differing convergence criteria, the unified methodology offers a framework for future investigation into mixed products of sine and cosine differences.