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- Thread starter spideyjj
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- Thread starter
- #1

- Mar 23, 2019

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$F_{1}(s)=\int_{-\infty}^{+\infty}e^{-i2\pi st}dt=0$ ( odd function ). And for s=0

$F_{1}(0)=\int_{-\infty}^{+\infty}1dt=+\infty$ so

$F_{1}$ is then defined by $F_{1}(s)=0$ if $s\neq0$ and $F_{1}(0)=+\infty$. $F_{1}$ is the Dirac delta "function" ( It's a distribution ). $F_{1}(s)=\delta(s)$

Now if $f(t)= cos(2\pi \omega t)$ then $f(t)=\frac{1}{2}(e^{i2\pi \omega t}+e^{-i2\pi\omega t})$. The Fourier transform of f is then $F(s)= \frac{1}{2}\int_{-\infty}^{+\infty}e^{-i2\pi (s-\omega)t}dt+ \frac{1}{2}\int_{-\infty}^{+\infty}e^{-i2\pi (s+\omega)t}dt= \frac{1}{2}F_{1}(s-\omega)+ \frac{1}{2}F_{1}(s+\omega)= \frac{1}{2}\delta(s-\omega)+ \frac{1}{2}\delta(s+\omega)$