Question

Using the generalized Plancherel identity, calculate the integral $$ \int_{0}^{\infty} \frac{d t}{\left(a^{2}+t^{2}\right)\left(b^{2}+t^{2}\right)}, \quad a, b>0 . $$

Short answer

\( \frac{\pi(b-a)}{2ab(b+a)} \)

Step-by-step solution

Understand the Integral's Structure

We need to calculate the integral \( \int_{0}^{\infty} \frac{dt}{(a^2+t^2)(b^2+t^2)} \). The integral involves rational functions with two quadratic terms in the denominator.

Use Partial Fraction Decomposition

Express \( \frac{1}{(a^2+t^2)(b^2+t^2)} \) as a sum of partial fractions: \( \frac{A}{a^2+t^2} + \frac{B}{b^2+t^2} \), and solve for \( A \) and \( B \).

Solve for Constants A and B

Set \( 1 = A(b^2+t^2) + B(a^2+t^2) \). By matching coefficients, solve for \( A = \frac{1}{b^2-a^2} \) and \( B = -\frac{1}{b^2-a^2} \).

Integrate Each Term Separately

The integral becomes two separate integrals: \( \int_0^\infty \frac{\frac{1}{b^2-a^2}}{a^2+t^2} dt - \int_0^\infty \frac{\frac{1}{b^2-a^2}}{b^2+t^2} dt \).

Use Standard Integral Form

Recall the standard integral \( \int_0^\infty \frac{dt}{t^2+x^2} = \frac{\pi}{2x} \). Apply this formula to each term: the first becomes \( \frac{\pi}{2a} \frac{1}{b^2-a^2} \) and the second \( \frac{\pi}{2b} \frac{1}{b^2-a^2} \).

Simplify the Expression

Combine the two results: \( \frac{\pi}{2(b^2-a^2)} \left( \frac{1}{a} - \frac{1}{b} \right) \).

Final Result

Simplifying further, we obtain the final result: \( \frac{\pi(b-a)}{2ab(b+a)} \).

Key concepts

Plancherel Identity

The Plancherel identity is a deep result in Fourier analysis that connects the field of signal processing to the mathematical study of functions. It essentially states that the Fourier transform is an energy-preserving transformation. This means that the total energy of a signal is identical whether the signal is measured in the time domain or in the frequency domain.
This identity is formally expressed as:

• The integral of the square of a function equals the integral of the square of its Fourier transform.
• This expression implies that certain calculations, such as integrals of rational functions, might be solved more easily by transforming them.

For the provided exercise, while the Plancherel identity isn't directly used to calculate the integral, its close relation to the Fourier transform shows that there are often alternative pathways in tackling complex integrals. Understanding these connections can deepen one's insight into calculus and the relationship between time and frequency representations.

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to simplify the integration of rational functions. For a complex fraction, it's broken into simpler fractions, which can be more straightforward to integrate separately.
In the exercise, the fraction \( \frac{1}{(a^2+t^2)(b^2+t^2)} \) is decomposed into two simpler fractions: \( \frac{A}{a^2+t^2} + \frac{B}{b^2+t^2} \). Here, the constants \( A \) and \( B \) are chosen to satisfy:

• Make the equality hold across the whole expression.
• This step makes integration easier for each part separately.

By solving the system of equations resulting from equating coefficients, we can determine \( A \) and \( B \), transforming a difficult integral into more manageable parts. This approach is crucial in integral calculus, particularly when dealing with polynomial denominators.

Integral Calculus

Integral calculus is fundamental in mathematics for finding quantities like areas under curves or solving differential equations.
In this exercise, integral calculus provides methods to approach and solve the integration problem by using different techniques and transformations:

• The given integral is initially transformed with partial fractions.
• Each part is then tackled using known forms and identities.

This divide-and-conquer methodology is often a key part of successfully solving such problems. By transforming the problem into simpler parts, we gain the ability to apply known solutions and formulas, easing the path to the final answer. The exercise highlights the step-by-step process through which complex integrals are broken down to reach an elegant and simplified solution.

Standard Integral Forms

When facing integrals, it's beneficial to know standard integral forms as they serve as shortcuts to quick solutions. These are predefined solutions for common types of integral problems.
In the exercise, the standard integral used is:

• \( \int_0^\infty \frac{dt}{t^2+x^2} = \frac{\pi}{2x} \)

This form is particularly useful when dealing with quadratic terms in the denominator, as it provides a direct solution to certain integrals without needing full derivation each time. By recognizing the structure of the integral and matching it to a standard form, the path to finding the solution is streamlined significantly, saving time and effort while ensuring accuracy. Understanding and memorizing these forms can greatly enhance one's problem-solving toolkit in calculus.