Fourier Series and Fourier Transform: A Brief Introduction

Fourier (well, actually his students)

believed that any function whatsoever could be expressed as a linear combination of sines and cosines of various frequencies.

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So then

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(An implicit assumption baked in: f(x) is a function with period 2π ^^)

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Then how are those coefficients a_n, b_n determined??!!?!?

Those coefficients must clearly be related to the function f(x) we’re trying to represent??!?!​

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To find them, multiply both sides of the above equation by sin nx or cos nx

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and use this relation.

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In a word, we’ll be using Fourier’s trick.

Where have I used this before — http://gdpresent.blog.me/220183169967

Electromagnetism I studied #8. Separation of variables

Previously we learned the method of images, which is one (cheeky) way to solve Laplace’s equation. Well, if you can call it a skill …

blog.naver.com

I’ve used it in electromagnetism^^hehehehe Well, is separation of variables only used in electromagnetism?

It’s used all over physics in general hehehe

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I’ll multiply both sides by cosnx.

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Then if we take the one-period average on both sides

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hahahaha

Why this is a trick —

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except for the blue one, everything flies away.​

In the end

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Each coefficient can be determined like this hehehe hehe

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In just the same way, get b_n and you get this~

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Then does the function always have to be one whose interval is from minus pi to pi???!?!!

Huh???? We can also just shift the interval and represent the function over that

The coefficients in that case are​

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​we can just write it like this, I guess

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But, both sine and cosine are exponentials by Euler’s formula??​?!!​

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That is,

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this equation, equivalently,

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can also be written like this.

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Now, to determine c_n, multiply both sides by

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We get this,​​

and if we take the -π ~ π average of both sides​

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Therefore in the above both-sides average, only the c_n term’s gotta survive….​

<To leave only c_n, we multiplied both sides by

, and to leave only c_-n, we’d multiply both sides by

.>​

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Also, as with sine and cosine, the period doesn’t have to be -π ~ -π, it can also be 0 ~ 2π, so​

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Like this, with period 2pi,

we’ve confirmed that f(x) on an interval like -π ~ -π or 0 ~ 2π​ can be expressed as a linear combination of other wave functions,

but then, does the period have to be 2π?!?!?!?!?!​​

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That’s no no.

Setting the period to 2l,

the logic that any~~ arbitrary f(x) can also be expressed!!!! means

we can represent any interval, any function.​​

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I’ll transform this formula-but-not-quite-formula thing.​

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Now we’re invincible.​

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Nope~. We need to send l to infinity to be truly invincible.

Then we can include even functions with ’no’ period.

So I’ll transform that formula again.​

So to jump to the conclusion

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Now how does this come out —

let’s take a slow~~~~ look. ​

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First, in this equation, let’s set nπ/l as α_n.

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And in that alpha, since n changes by 1 each time,

​α_n changes by π/l.

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From here it’s less math and more of a hack​​

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​First, let’s call Δα as dα, we can do that right

Then the n in α_n doesn’t mean anything anymore. Let’s just write it as α.​

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And I’ll send l (ell) to infinity.

Then the n in the sigma out front, let’s just write it as an integral!!!!​

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So we can write it like this, let me play around a bit.​

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I’ll stick the red part into those parentheses.​

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Con to the clusion​

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Gonna use this~~~~~~~~~~~~~~~~~~~~~~

It’s called the Fourier transformation,

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and the meaning is​

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whether you see the domain of some wave as a linear combination of waves with various frequencies

that is, whether you view the domain in terms of frequency or in terms of time

it’s an equation with that kind of meaning.​lollollol

Originally written in Korean on my Naver blog (2016-07). Translated to English for gdpark.blog.