M2023 - Fourier Series : 3 - arith. freq + geo. amp =? circle

by using fourier series, we can create a lot of periodic waves. here are some of the famous, important waves for examples. In this case, although the final trace on the 2D complex plane doesn't resemble any waves, but we can keep track of the final Y-position of the final vector v[t]

sawtooth :

square :

here the amplitudes are decreasing as the denominator is arithmatically increasing.

but what if the amplitude is decreaseing as the denominator is geomatrically increasing ?

in other words, here's an example :

if we don't care about the green trail (Y-position) for now, we can see that the final white trail v[t] traces ... a circle ?

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I tried with other shrinking ratios, and we ended up with different circles.

Let's rewrite the whole thing in a more math friendlier way.

this means that the final vector v[t] should look something like this :

Notice that the sum in the parenthesis can be treated as a geometric series :

in order for the sum to converge, the magnitude of the ratio must be smaller than one

so the sum would be :

and we can continue from there :

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to proof that it is a circle, we will need to propose 2 things :

• there exists a center point for the circle

• the distance between v[t] and c will be constant, which is the radius of the circle

let's start with finding the left most and right most point of the "circle"

therefore the proposed center is :

and the proposed radius is :

now we have to prove that the following is true

and here is the proof that v[t] traces a circle :

QED (OwO)/

some of the important concepts here are

• complex numbers

• complex conjugate

• complex modulus