On How Physics Can Proof Math Expressions

Hello everyone,

Recently, while working on designing a clock distribution network, I encountered a task that requires evaluating the power contribution of the fundamental harmonic of a rectangular wave in relation to the overall power. To tackle this problem, I utilized Fourier analysis. However, what fascinated me the most and prompted me to share this case with you is the intriguing connection it had with the field of pure mathematics, particularly the Riemann's Zeta function. This presents an opportunity for you to test your intuition and take a guess of this ratio.

If you're not familiar with the fact that infinite sums often involve π, I encourage you to watch the enlightening video by 3Blue1Brown, which provides excellent mathematical insights using engaging infographics. In addition, you can read on Basel Problem.

Applying Fourier analysis, I swiftly determined that the Fourier series coefficients of an odd unity rectangular wave are given by [1]. From a power perspective, these coefficients need to be squared. Therefore, the ratio I look for can be expressed as [2]. Since all the even coefficients are zero, a simpler expression emerges [3].

At this point, I was clueless how to evaluate this infinite sum. As an engineer, I opted to utilize an Excel spreadsheet to compute it for me, up to the 1000th term. In case you're curious, the answer is approximately 81%, which resonated with my intuition. I hope your guess wasn't too far off either.

However, a few hours later, I became determined to understand the origin of this ratio—why is it 0.81056946913870217155103570567782?! Then I recalled that the root mean square (RMS) value of a sine wave is the amplitude divided by √2. For a unity rectangular wave, the RMS value is simply 1. Therefore, from an RMS perspective, which is also related to power, our ratio is given by [4]. In case you're wondering, this fraction is our 0.81...

This discovery not only pleased me because I succeeded in understanding the origin of this ratio, but also because I found it impressive that a basic engineering approach allowed me to solve a pure mathematics problem. If you search the web for the infinite sum of odd inverse squared integers [3], you will come across complex developments involving the Gamma function, Taylor series approximations of trigonometric functions, complex numbers, and calculus. As you just read, I found a solution to this pure math problem using fundamental engineering knowledge.

For me, uncovering instances where pure mathematics and engineering intersect in such a harmonious manner can truly make my day.

This is my first article on LinkedIn. I hope you found this interesting as I do. Please share your thoughts in the comments section.