Answer to Question #85505 in Complex Analysis for Shiva

Question

Determine the Fourier transform of the function
f(t)=1-t. 0 ≤t≤1
1+t. -1≤t≤0
0. Otherwise

Answer 1

$$\hat{f}(k)=\int\limits_{\mathbb{R}}f(t)e^{-2\pi i kt} \, dt$$

$$=\int\limits_{-1}^0 (1+t) e^{-2\pi i kt} \, dt+\int\limits_0^1 (1-t) e^{-2\pi i kt} \, dt$$

$$=\dfrac{2\pi i k-e^{2\pi i k}+1}{4\pi^2 k^2}+\dfrac{2\pi i k+e^{2\pi i k}-1}{4\pi^2 k^2}$$

$$=\dfrac{\sin^2(\pi k)}{\pi^2 k^2}$$