The Fourier transform of the linear function, represented by the integral \(\int_{-\infty}^{+\infty} x e^{ikx}\, dx\), results in the expression \(-2\pi i \frac{d}{dk} \delta(k)\). This derivation involves recognizing that the integral is intrinsically undefined, and the presence of the minus sign is necessary to maintain the integrity of the original integral. The discussion highlights the relationship between differentiation and the Dirac delta function in the context of Fourier transforms.
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Hmm.. seems to make sense. Why is there a minus sign popping up?vanhees71 said:This expression doesn't make sense since it's intrinsically undefined. A handwaving way is
\int_{\mathbb{R}} \mathrm{d} x x \exp(\mathrm{i} k x)=-\mathrm{i} \frac{\mathrm{d}}{\mathrm{d} k} \int_{\mathbb{R}} \mathrm{d} x \exp(\mathrm{i} k x)=-2 \pi \mathrm{i} \frac{\mathrm{d}}{\mathrm{d} k} \delta(k).