why does cos(wt) disapear?

[low belfry]

Im trying to figure out the fourier transform of g(t) and i'm almost there. I do get every part of the solution but just not how the cos term disapears.

That is, since our function is only defined on -1,1 we can just integrate over that rather then -inf to inf. And we use eulers identity on the exp term. And after that we have an odd function times an odd, hence 2 times over the half interval.

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[sage arrow]

Becuase g(t) is odd. As you can see, that's also why we multiply the sine term by 2.

[low belfry]

I dont get it sorry 🤔

that makes the even part disapear?

[sage arrow]

Suppose we have a convergent ∫(f(x)dx, -a, a). Then:

1. If f(x) is even, then ∫(f(x)dx, -a, a) = 2∫(f(x)dx, 0, a).
2. If f(x) is odd, then ∫(f(x)dx, -a, a) = 0.

Moreover, the product of two even or two odd functions is even, and the product of an even and an odd function is odd.

[low belfry]

yeah

but for our (cos(wt) - i sin(wt))

What made the cos part go away?

im so lost rn lol

[sage arrow]

Cosine is an even function, sine is an odd function.

[low belfry]

yeah

[sage arrow]

Yeah.

[low belfry]

Hmmmm

oh wait

did they split it up

and then cos over -1, 1

= 0?

naah

[sage arrow]

g(t) is odd. So:
∫(e^(-iωt)g(t)dt, ℝ) = ∫(e^(-iωt)g(t)dt, -1, 1) = ∫((cos(ωt)g(t) - i sin(ωt)g(t))dt, -1, 1) = ∫(cos(ωt)g(t)dt, -1, 1) - i∫(sin(ωt)g(t)dt, -1, 1)
cos(ωt)g(t) is odd and sin(ωt)g(t) is even, so:
∫(e^(iωt)g(t)dt, ℝ) = 0 - 2i∫(sin(ωt)g(t)dt, 0, 1) = -2i∫(sin(ωt)g(t)dt, 0, 1)
The rest is easy.

Ah, sorry, hold on.

Misread the function, one sec.

[low belfry]

haha np!

[sage arrow]

Alright, done.

[low belfry]

okay i will try to understand now^^

OH YEAH

[sage arrow]

A consequence of this: if g(t) is an even function, then the Fourier transform will only contain cosines and will be purely real. And if g(t) is odd, then its Fourier transform will only contain sines and will be purely imaginary, like in your case.

[low belfry]

YEAH

nooow i get it

i was there at the cos part

just that i forgot g(t) is odd

or like mulitplied with them both

and then yes it became 0!

[sage arrow]

Yeah. Just not 0! = 1, though 😄

[low belfry]

xd

tysm!

+close