I'm trying to find out by myself the Fourier Series of a Dirac Train, but I'm getting after Integration by Parts that Dn equals to 0 and not to 1 as needed to be. Could you please help me find my mistake?
So, given:
$x(t)$$=$$\Sigma$$Dn$$e$$^{i{\omega}nt}$
Whilst:
$x(t)$$=$$\Sigma$$\delta$$(n)$
I'm finding out the $Dn$ (I cut some obvious equations here):
$Dn=1/P{\int}{\delta}(n)e^{-i{\omega}nt}=$
And now I'm getting to trouble...
Here I do integration by parts.
From:
${\int}f'(t)g(t)=f(t)g(t) - {\int}f(t)g'(t)$
We get the following, by assuming that $f'(t)={\delta}(t)$, which gives us that $f(t)={\int}{\delta}(t)=1$ :
${\int}{\delta}(n){e}^{-i{\omega}{n}{t}} = 1*{e^{-i{\omega}nt}}-{\int}{1*-i{\omega}nt*e^{-i{\omega}nt}} = 1*{e^{-i{\omega}nt}}+i{\omega}nt*{\int}e^{-i{\omega}nt} = 1*{e^{-i{\omega}nt}}-i{\omega}nt/i{\omega}nt*{e^{-i{\omega}nt}}= 1*{e^{-i{\omega}nt}} - 1*{e^{-i{\omega}nt}} = 0$
Where is my mistake?