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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 |
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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. |
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