Uncovering the Hidden Power of Fourier Series

[matqkks]

If we have a simple periodic function (square wave) which can be easily written but the Fourier series is an infinite series of sines and cosines. Why bother with this format when we can quite easily deal with the given periodic function? What is the whole point of dealing this long calculation?

 

[HallsofIvy]

What makes you think "we can quite easily deal with the given periodic function"? Can you give an example?

 

[matqkks]

I meant to say why write this as an infinite series when it can be expressed quite easily as a waveform. For Maclaurin or Taylor series it makes it easier to handle some difficult functions such as sin(sqrt(x)).

 

[Stephen Tashi]

If all one intends to do with a function is write it down, then it might be simpler to write the function as a rule which has some "if ...then..." conditions in it than write it as an infinite series. For example, a square wave (as a function of time alone) could be written in a format like: if ( n < t < n+1 and n is an even integer) then f(t) = 1. Otherwise f(t) = 0. If you need to integrate or differentiate a function, the "if...then..." conditions can be a nuisance and it may simpler to deal with the infinite series.

A true square wave isn't differentiable at the jumps. In a situation (such as in electronics) when we are dealing with a nominal square wave, we could make a realistic model for the nominal square wave by using some "if...then..." conditions to round the shape of the jumps. However, it may be simpler to think of the square wave as an infinite series and then neglect some of the terms of the series in order to achieve the same sort of approximation.

The "response" of some physical systems to an "input" (e.g. the effect of a electronic filter on an input signal) may be simple to analyze when the input is a sine or cosine function. For a linear system, the response to the sum of inputs is the sum of the responses to the individual inputs. Hence the simplest analysis is often to represent an input signal as a sum of sine and cosine functions and compute the response as the sum of the individual responses.

 

[homeomorphic]

Maybe one day, you'll be abducted by aliens who will ask you to solve a "cooling off" problem for a thermally isolated rod.

And then you'd be able to answer them because you know how to solve the problem when the initial conditions are sinusoidal, those being the eigenfunctions of the problem (they die off exponentially because the heat equation doesn't like steep temperature gradients, so it kills them off rapidly), so if you can express the arbitrary initial conditions as a Fourier series, you'd be good to go.