Fourier transform - what integral limits

[zezima1]

Find the Fourier transform of the unit rectangular distribution f(t) = 1 for ltl<1 else 0
Since e^-iωt is zero except for t in ]-1;1[ it must be an integral over this interval. But should I take the boundaries as -1 and 1? Because they are not included in the interval where e^-iωt is not zero but rather supremum and infimum for it. Would it better to put the limits lim t->1 and lim t->-1?

 

[MathematicalPhysicist]

What?
The exponent never vanishes for finite values.

 

[zezima1]

umm the Fourier transform is:

∫_-∞^∞f(t)e^-iωtdt

But since f(t)=0 for t≥1, t≤-1 you integrate from -1 to 1?

 

[LCKurtz]

umm the Fourier transform is:

∫_-∞^∞f(t)e^-iωtdt

But since f(t)=0 for t≥1, t≤-1 you integrate from -1 to 1?

Yes.

 

[vela]

Find the Fourier transform of the unit rectangular distribution f(t) = 1 for ltl<1 else 0
Since e^-iωt is zero except for t in ]-1;1[ it must be an integral over this interval. But should I take the boundaries as -1 and 1? Because they are not included in the interval where e^-iωt is not zero but rather supremum and infimum for it. Would it better to put the limits lim t->1 and lim t->-1?

I think you meant f(t) where you wrote e^-iωt.

Regarding your question about the limits, it turns out it doesn't matter. You can look up some statement about the uniqueness of Fourier transforms. I think it's something like two functions have the same Fourier transform if they differ over a set of points of measure 0.