During my research this integral has shown up


$ \frac{1}{2T} \int\_{-T}^T \left( 1 - \frac{|\tau|}{T}\right)e^{-\alpha\tau^2}\cos(2\pi f\_0 \tau) d\tau$


I tried to solved by taking the real part of a the complex exponential but it didn't work. Any help?


Cheers,


Mikitov



Integral and limit
The limit is 0, the integral from $0$ to $T$ is:
$$\frac{i \sqrt{\pi } e^{-\frac{\pi ^2 f^2}{a}}
 \left(\text{erfi}\left(\frac{\pi f-i a
 t}{\sqrt{a}}\right)-\text{erfi}\left(\frac{\pi f+i a
 t}{\sqrt{a}}\right)\right)}{4 a^{1/2}}
$$
$$
-\frac{\pi ^{3/2} f e^{-\frac{\pi ^2
 f^2}{a}} \text{erfi}\left(\frac{\pi f-i a t}{\sqrt{a}}\right)+\pi
 ^{3/2} f e^{-\frac{\pi ^2 f^2}{a}} \text{erfi}\left(\frac{\pi f+i
 a t}{\sqrt{a}}\right)-2 \pi ^{3/2} f e^{-\frac{\pi ^2 f^2}{a}}
 \text{erfi}\left(\frac{\pi f}{\sqrt{a}}\right)}{4ta^{3/2}}
$$
$$
+\frac{\left(1+e^{4 i \pi f t}\right) e^{-t (a t+2 i \pi f)}-2}{4ta},
$$
as per Mathematica.


