1. ## Help!!! Fourier

My def. of FT is
$\displaystyle \int_{-\infty}^{\infty}f(t)e^{\frac{i2\pi t}{T}}dt$, T= period

* How to calculate Fourier transform for
$\displaystyle f1(t)= e^{-t},$ for $\displaystyle t >= 0$; and $\displaystyle = e^t$ for $\displaystyle t<0$ (Which is the period here???)

*$\displaystyle f2(t)=2,$ for $\displaystyle -1/2<t<1/2$, this gives me $\displaystyle \frac{-e^{\alpha t}}{\alpha} |_{-\infty}^{\infty}=\infty$ I don't remember what to do with that...
Maple says that FT in this case is $\displaystyle 2\pi\delta(\pi)$.

2. Originally Posted by Quantum_Alpha
My def. of FT is
$\displaystyle \int_{-\infty}^{\infty}f(t)e^{\frac{i2\pi t}{T}}dt$, T= period

* How to calculate Fourier transform for
$\displaystyle f1(t)= e^{-t},$ for $\displaystyle t >= 0$; and $\displaystyle = e^t$ for $\displaystyle t<0$ (Which is the period here???)

*$\displaystyle f2(t)=2,$ for $\displaystyle -1/2<t<1/2$, this gives me $\displaystyle \frac{-e^{\alpha t}}{\alpha} |_{-\infty}^{\infty}=\infty$ I don't remember what to do with that...
Maple says that FT in this case is $\displaystyle 2\pi\delta(\pi)$.
This gives me the impression that you are confusing Fourier series with Fourier transforms.

If a function is periodic, with period T, then it has a Fourier series, in which the n-th (complex) Fourier coefficient is $\displaystyle \hat{f}(n) = \frac1{T}\int_0^Tf(t)e^{i2\pi t/T}dt$.

Fourier transforms are for functions (non-periodic) that are integrable over the whole real line, and the Fourier transform is a function, defined by $\displaystyle \hat{f}(w) = \int_{\infty}^{\infty}f(t)e^{-iwt}dt$.

For the two functions f_1 and f_2, the integrals for the Fourier transforms are
$\displaystyle \hat{f}_1(w) = \int_{-\infty}^0e^{t}e^{-iwt}dt + \int_0^{\infty}e^{-t}e^{-iwt}dt$,
$\displaystyle \hat{f}_2(w) = 2\int_{-1/2}^{1/2}e^{-iwt}dt$.

3. Ok, I will resolve these integrals, I hope not to have problems with the $\displaystyle \infty$'s

4. Gives me $\displaystyle \hat{f}_1(w)=\frac{2}{1-iw}$, and $\displaystyle \hat{f}_2(w) = -\frac{2}{iw}(e^{\frac{-iw}{2}}-e^{\frac{iw}{2}})$.
Are they ok?

(Oh! About period, it was because in my def. I have $\displaystyle w =2\pi/T$)