Calculating Frullani Integrals Using Series Expansion: A Detailed Guide

A Frullani integral is an integral of the form (int_{0}^{infty} frac{f(ax) - f(bx)}{x} , dx), where (f) is a continuous function. This article explores the process of calculating such integrals using series expansion, with a specific example. We delve into the methods and provide detailed proofs and explanations for better understanding.

First, let's consider a function (f(x)) defined by the alternating power series:

(f(x) sum_{k0}^{infty} frac{(-1)^k}{k^3} x^k)

Using this series, we aim to calculate the Frullani integral:

(int_{0}^{infty} frac{f(2x) - f(3x)}{x} , dx)

Step 1: Series Expansion of (f(x))

We start with the given series for (f(x)): (f(x) sum_{k0}^{infty} frac{x^k}{k^3})

By manipulating the series, we can rewrite (f(x)) as:

Multiply both sides of the series by (x^2): (x^2 f(x) sum_{k0}^{infty} frac{x^{k 2}}{k^3}) Integrate both sides from 0 to (x):(int_{0}^{x} x^2 f(t) , dt int_{0}^{x} sum_{k0}^{infty} frac{t^{k 2}}{k^3} , dt -frac{x^2}{2} - log(1-x) - x)

From this, we can deduce that:

(f(x) sum_{k0}^{infty} frac{x^k}{k^3} -frac{1}{2x} - frac{1}{x^2} - frac{log(1-x)}{x^3})

As a result, the behavior of (f(x)) is characterized by a vertical asymptote at (x1), with (lim_{x to 1^{-1}} f(x) infty).

Step 2: Transforming the Function for Proper Integration

To simplify the calculation of the Frullani integral, we need to ensure that the integral converges. This can be achieved by reflecting the function about the (y)-axis. Define a new function:

(f(x) frac{1}{2x} - frac{1}{x^2} - frac{log(1-x)}{x^3})

Then, we reflect it about the vertical axis to get:

(f(-x) frac{1}{2x} - frac{1}{x^2} frac{log(1 x)}{x^3})

Now, (f(-x)) satisfies the conditions for the Frullani integral, specifically:

The function has a finite limit at (x0): (lim_{x to 0} f(-x) frac{1}{3}) The function has a finite limit at (x to infty): (lim_{x to infty} f(-x) 0)

With these properties, we can now calculate the Frullani integral:

(int_{0}^{infty} frac{f(2x) - f(3x)}{x} , dx left[f(0) - f(infty)right] log left(frac{3}{2}right) frac{1}{3} log left(frac{3}{2}right))

Conclusion and Further Resources

This approach showcases the importance of series expansion and reflection properties in solving Frullani integrals. For more information and problem-solving tips in mathematics, physics, and computer science, visit the PROBLEM Lemma YouTube channel.