Some dude asked to evaluate the following integral: $$\int_0^{\infty} dt \, \frac{\sin^3{\pi t}}{(\pi t)^3} \cos{u t}$$ I propose to perform a direct computation using Cauchy’s theorem, i.e., extension into the complex plane. I will then verify the solution using the convolution theorem. DIRECT EVALUATION Rewrite the integral as $$\frac12 \int_{\infty}^{\infty} dt \, \frac{\sin^3{\pi t}}{(\pi t)^3} […]

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