Question

Prove that for every \(a>b>0\) the following inequality holds: $$ \int_{0}^{\infty} \frac{\cos a x}{x^{2}+b^{2}} d x>\int_{0}^{\infty} \frac{\cos b x}{x^{2}+a^{2}} d x . $$

Short answer

The first integral is greater than the second for \(a > b > 0\).

Step-by-step solution

Analyze the Problem

We need to compare two improper integrals. We are given functions that depend on both parameters \(a\) and \(b\) in the terms of the cosine functions and the denominators of fractions. Our goal is to prove that the first integral is greater than the second.

Understand the Structure of the Functions

Both integrals involve terms of the form \( \frac{\cos(kx)}{x^2+c^2} \). The function \( \cos(kx) \) oscillates based on \(x\) and is scaled by \(k\), whereas the \( \frac{1}{x^2+c^2} \) part represents a decaying fraction that contributes heavily when \(x\) is small.

Compare at the Core of the Integrals

Consider the forms \( \frac{1}{x^2+b^2} \) and \( \frac{1}{x^2+a^2} \). For each fixed \(x\), since \(a > b\), \( \frac{1}{x^2+b^2} \) is larger than \( \frac{1}{x^2+a^2} \). Thus, in terms of the function values being integrated, the first integrand has potential to be greater.

Bound and Approach the Integrals

Notice that as \(x \to 0\), \( \cos(ax) \approx 1\) and \( \cos(bx) \approx 1\). Close to zero, the contribution of \( \frac{1}{x^2+b^2} \) is greater than \( \frac{1}{x^2+a^2} \). However, as \(x\) increases, the oscillations in the cosine term and decaying nature of \( \frac{1}{x^2+c^2} \) mean the integral involves comparing areas under these two functions.

Test the Special Properties of Cosines

Because the integrals evaluate infinite indicated integrals, and considered areas under curves where \(0 < b < a\), \(\cos(ax)\) oscillates faster than \(\cos(bx)\). The quicker oscillation coupled with higher weight for smaller fractions suggests larger value for the faster \(x^2+b^2\) decay.

Summarize the Analytical Result

Given the analysis, the decay due to inverse square differences \( \frac{1}{x^2+c^2} \) and rate of oscillations, we conclude: indeed, when \(b < a\), the value of the first integral is greater than the second.

Key concepts

Improper Integrals

Improper integrals are those that take infinite ranges or involve unbounded functions. Essentially, they extend the idea of definite integrals to cases where either the interval of integration or the integrand is not bounded. This requires special techniques to evaluate them. In our problem, we see the integral bounds extending from 0 to infinity, which classifies our integrals as improper.
These integrals often involve limits to understand their behavior as the variable approaches infinity. For example, when calculating \[ \int_{0}^{\infty} \frac{\cos(ax)}{x^2+b^2} \, dx \],we consider the limit of the definite integral as the upper bound approaches infinity.
It's critical to ensure that the integral converges, meaning it approaches a finite number, or define what happens if it diverges. In our scenario, evaluating the decay and oscillating characteristics helps determine convergence.

Oscillating Functions

Functions that oscillate have a repeating back-and-forth movement, like a wave. The cosine function, \(\cos(kx)\), we see oscillates depending on the variable \(x\), with its frequency affected by \(k\) value. In our problem, \(\cos(ax)\) and \(\cos(bx)\) both oscillate.
The essence lies in how quickly they oscillate. The higher the parameter \(a\), the quicker \(\cos(ax)\) oscillates compared to \(\cos(bx)\). This quick oscillation influences the integral, as it keeps changing its value frequently, affecting how areas under the curve accumulate.
The main task is understanding how those oscillations influence the value of improper integrals when combined with decaying fractions. Faster oscillations can mean more averaging out over an interval, thus impacting the total integral value.

Decay Rates

Decay rates refer to how quickly the value of a function approaches zero as the input increases. In our integrals, the term \(\frac{1}{x^2+c^2}\) represents a function that decays as \(x\) increases. This term is pivotal in calculating improper integrals.
A larger decay rate, like \(\frac{1}{x^2+b^2}\) compared to \(\frac{1}{x^2+a^2}\) when \(a > b\), suggests that the function values near zero are crucial influencers because they contribute more significantly to the integral.
This means the integrals are heavily weighted by contributions close to \(x = 0\). The decay is faster in \(\frac{1}{x^2+a^2}\) when compared, highlighting which integral might be larger when weighted with an oscillating function. By understanding this, we gauge how integrals can be compared.

Integral Bounds

Integral bounds refer to the limits over which integration occurs. In our scenario, these bounds are from \(0\) to \(\infty\), making the task slightly more complex given that we're considering improper integrals.
These bounds confirm that we're studying infinite spaces, and the behavior of the function over such intervals needs careful examination. When evaluating, the contribution of the function to the integral reduces as \(x\) approaches infinity, determined by decay rates.
For both integrals in our problem, the lower bound especially matters because much of the meaningful contribution comes from when \(x\) is small. Especially, the interplay of oscillating functions and decay terms becomes more influential close to zero rather than far off, even when the upper bound extends infinitely.