Unraveling the Intricacies of the Fourier Series
A Journey Through the Mathematical and Philosophical Insights
What is the Fourier series?
The Fourier series is a powerful mathematical tool for decomposing periodic functions into a combination of sine and cosine functions. It allows for a comprehensive analysis of complex waveforms.
When was the Fourier series first announced?
Joseph Fourier, a pioneer in his field, first announced the Fourier series in 1822. His groundbreaking work initially aimed to study heat flow in solid objects.
What happens when a class teacher tries to teach the Fourier series in a classroom?
The teacher begins writing on the board from the top left corner, trying to derive some equations. When he reaches the bottom left corner, he realises he has run out of space and needs to erase everything and start over on another part of the board. In the meantime, one hour has elapsed, class is over, and the teacher says, "We will continue tomorrow." Everyone sighs and rushes out of the classroom.
The next day, the teacher tries again, starting with the critical questions he would set in the exams. The students try to be more engaged and interact with the lesson, eager to memorise the material for the upcoming test. Yet, as the class progresses, they struggle to stay focused and maintain their energy levels. Somehow, the semester ends, and the curriculum moves forward, leaving behind the complex Fourier series.
Little did the teacher and the students know that the Fourier series would continue to haunt them forever.
From 1822 until the late 1990s, when Excel programming evolved, the Fourier series remained among the most tedious chapters in math. Even after that, students and teachers struggled to grasp the concept entirely.
No matter how diligently they try to decipher his work, it is extremely difficult for a twenty-year-old to comprehend Fourier’s ideas fully. When the person reaches fifty-something or retires from a profession, he will connect the dots to what he could not have done thirty years earlier. Over time, a profound realisation dawns about the intricate nature of the Fourier series, shedding light on its once daunting aspects.
As daunting as it may seem, the Fourier series is nothing new. It is the same knowledge that sages in Vedic times have been preaching and practising. Ancient sages in Vedic times had an innate understanding of the concepts Fourier enunciated in 1822, demonstrating a seamless transmission of knowledge across generations without the need for formal assessments.
Fourier had no option but to use the sine function in his derivations. A sine function is a circular function. A circle is characterised by being a closed surface, implying stability. One constant, pie, governs all circles and closed surfaces in the universe. This explains periodicity, probability, distribution, predictability, interpolation, extrapolation, and all other physical phenomena in nature. The often-repeated phrase "everything is planned" is derived from this notion.
When you introspect, you will realise every event in one's life is predetermined. Any given event is subject to N variables that ultimately determine the outcome. Each variable varies in amplitude, direction, and frequency. This shapes the course of an event, leading to the conclusion that everything is planned. No factor, however minuscule, can be denied its influence on the result. Every action, decision, and circumstance play a role in determining the outcome. This is the essence of the Fourier series.
In the traditional Gurukul, the guru advises his pupils to discern valuable lessons, discard negative influences, and integrate the wisdom gained to successfully navigate life's challenges.
The guru says, "See who you are and where you have come from, your family background, and the time and place of your birth. You are a son, a grandson, a brother, a father, a husband, a neighbour, a citizen, and a community member. You may be a friend to someone, a helper to others, or a leader to others, and you may have enemies, too. You may have pets, too. You have responsibilities. Each of these requires your time, attention, and energy. You are torn about understanding the vicissitudes. See how these forces influence your actions."
"You may determine your karma, but you have no control over your birth. That is your destiny. It is irrevocably attached to your parents."
"You eventually act according to the most dominant of these forces. But you can not deny that the least dominant did not affect you somehow. You considered them, even if they did not sway your ultimate decision."
"You then realise that these forces predetermine every incident in your life, your fate shaping the person you become. This exemplifies the cyclic nature of a circle, showcasing its perpetual continuity without distinct origins or conclusions. It is a continuous cycle of ups and downs, depicted in the sine wave as crests and troughs. What goes around comes around."
"You lie somewhere in this circle. You are made of five elements of nature and will eventually dissolve into them. Matter, or energy from matter, can neither be created nor destroyed; it can only be transformed."
"What you see is the result of multiple forces and elements working together, not just one or more forces. It's a combination of forces and elements working together. A book lying on the table is not only under the influence of the Earth's gravity but also of the electromagnetic forces between its particles and extra-terrestrial influences, just like anything else. When you see a successful person, it is not only because of their efforts. Success is not just about individual effort but also about the support and sacrifices of others. It is the culmination of various forces and elements working together. The next time you see someone succeeding or failing, remember that their success or failure is not solely their own but a collective effort of many different factors working in unison."
The Fourier series applies to a function that determines such harmonics as a sum of sine and cosine functions. It implicitly accepts that every straight line, however long, is an arc of a circle. A straight line has two points as endpoints. For stability to be attained, a surface must be enclosed, emphasising the critical role of closed structures in maintaining equilibrium and reliability in systems. Two intersecting lines represent an open, unstable system; you need a third line to complete a triangle and make the system stable. A triangle is a closed loop; a circle is its broader manifest.