# FOURIER SERIES: Expresssing the alphabets of Mathematics

Everything in this world is changing, fastly or slowly, but surely. Being engineers the things that matter to change for us are the temperature in a space,  the voltage across any terminal, current in a loop, the electromagnetic field in space, the velocity of fluid flow, etc.

We can classify any change in physical quantity either as periodic or non-periodic. If the change in quantity repeats itself or follow some definite pattern in a finite time interval, then it is periodic else non-periodic.

Now why an engineer is interested in a change of magnitudes of such quantities? Basically, engineers stand for simplifying common man’s life. A man has to have control over the many physical quantities around him, controlled change in this quantities are proved to be useful for him. For example in communication, from the technical point of view, it is fine control of pressure difference in the air by vocal cords that cause the eardrum to vibrate and generate required nervous sensation which is felt by the brain. In electromagnetism change of one causes another to come into existence (many machines runs on this principle), etc.

To simplify our life we need machines to care take of as many things as possible. So we want reliability from them or we can depend on them, in other sense, we actually know how they are going to behave in future, that’s why we can depend on them, and this property inherently comes from the periodic change in physical quantities associated with these machines. Thus it becomes very important to know more about the nature of these periodic phenomena.

So these phenomena of changes are described by a set of rule, mathematically we call them the function, many of which are quite complex in nature. We want to be reliable and predictable with them also, so how can it be done. Like sound signal are described by weird functions, varied pressure changes.

The philosophy behind this mathematical concept is that anything in this world, however complex they may appear, at the root level fundamentally made of basic units. Eg in daily life – however wide variety of organs human, at microscopic level they are ultimately made of cells, however high and different a building is it is made up of bricks, however wonderful a novel of literary work is, it is made of different combinations of basic 26 alphabets only.

So by intuition, we can say that these elements of mathematical literature, the functions, must also be made up of the most fundamental periodic functions “the sine” and “the cosine”.

This is what exactly done in Fourier analysis.

The question is, HOW CAN WE USE SUCH SIMPLE FUNCTIONS TO MODEL VERY COMPLEX FUNCTIONS?

We will study the topic in the chronological order in which the subject got developed, i.e. understanding first the Fourier series and then its limiting case the Fourier transform for the non-periodic functions.

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