Simple integrals questions

[mortal rivet]

I think that the prof is doing something wrong, but hes probably right, thats my experience haha.

The question asks us to calculate the fourier series of f(t) with period = 1, but the tricky part is that its only defined on an interval.

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[mortal rivet]

He drawed it like this, how can we know that it is also "t" on the other intervals, that isnt said in the question?

And by the look of this then "t" is no longer an odd function, but rather even.

[buoyant flame]

Oh, usually the function is either assumed to be 0 or periodic.

[mortal rivet]

Oh

[buoyant flame]

As for the transform... Hm, I think you can set it as an infinite sum of integrals.

[mortal rivet]

aah yeah

here comes my other question

cause here is where i think he does something quiet weird

[mortal rivet]

also this is a tricky question in my opinion

cause u usally think "t", oh yeah an odd function

but then if u chek the intervals its not 💀

so its like a bait

or "booby trap"

[mortal rivet]

So this is the formula for the coefficents

[buoyant flame]

Those aren't the most general formulas.

[mortal rivet]

Yeah thats true

but its what he wants us to use on the test

so we'll have to go with that xd

[mortal rivet]

and this is what he wrote

Shouldnt it be 4 times (0, 1/2)?

cause T = 1

[buoyant flame]

Here are the formulas I'm used to using. The text reads "general series of f(x) for x ∈ (x0 - l, x0 + l)".

In our case x0 = 1/2, l = 1/2.

[mortal rivet]

ooh

yeaah i prefer those too

[buoyant flame]

I'm not sure if that will work. Rather, I think that will produce the following function.

[mortal rivet]

yeah i was gonna say cause

he ended up with an = 0

[buoyant flame]

Did your professor calculate the coefficients? We can just check what they give us.

[mortal rivet]

yeah

So he ended up with an = 0

but yet his a0 = 1?

[buoyant flame]

Ohh. I see what he did!

[buoyant flame]

Translate this function by 1/2 down. What will you get?

Try making a sketch.

[mortal rivet]

🤔

wdym exactly lol?

by 1/2 down, sorry.

[buoyant flame]

Move the function by 1/2 units down.

[mortal rivet]

well i get t/2?

[buoyant flame]

No, don't divide it by 2.

When you move y = f(x) down by 1/2, you get y = f(x) - 1/2.

[mortal rivet]

ahaa

oh yeah it becomes

symetrica around

x axis

[buoyant flame]

Well, kinda, but look at the origin.

The function now has a nice property.

[mortal rivet]

hmmm

im sketching it rn

oh yeah its shifted to the right x-axis

oh yeah he made it

odd?

@buoyant flame its "starts" at the origin?

oh yeah from 0,1

instead of 0, to 1/2

[buoyant flame]

Yeah!

Basically, our g(t) is a sum of 1/2 and an odd function.

[mortal rivet]

what the hell thats freakin brilliant

[buoyant flame]

So, a(0) will be 1, a(n > 0) will all be 0 and b(n) will have to be calculated.

[mortal rivet]

yeaaah

[buoyant flame]

Yeah, that's quite a good idea to do here. Reduces work by quite a lot.

[mortal rivet]

yeah

so he makes it odd

i wish he wrote that he did that atleast on the sheet

uh i understood it there but now im lost cause i took a break xd

would you like to help me visualize it again?

[buoyant flame]

Here is g(t) and g(t) - 1/2.

[mortal rivet]

ohh yeah

and since we do this

instead of 2 times (0,1/2)

we just do 2 times 0,1

i think im just tierd atp but i get how he solved it!

what im just left at is why he hasnt written it anywhere in his solution

or like how it comes "in"

This is what he lastly wrote and solved and thats the end.

[buoyant flame]

Yeah, I'm not sure that's a good thing to skip that step.

If you do, you need to use the general formula.

[mortal rivet]

Yeaah

like with the general formula it makes sense

but with ours it says instead from -1/2 to a half

and then ofc 2 times 0 to a half

but it still doesnt end up to 1

maybe i should skip this question since it has only showed up 1 time out of like 10 exams

during the past 8 years haha

[buoyant flame]

Well, you can just apply the logic we did above: shift the function so that it becomes odd, then find the coefficients, then shift it back.

[mortal rivet]

aah jeez

no pain no gain

i would probably learn it if i had the time

but i have 2 other exams aswell

[buoyant flame]

Oh. Well, then that's up to you.

Good luck!

[mortal rivet]

Thanks!

I've studied really hard for this so i think i'll pass!

i even have a whole sheet where i memorized a ton of stuff

[fickle trench]

@mortal rivet has given 1 rep to @buoyant flame

[mortal rivet]

+close