# Fourier transform of the linear function

## Main Question or Discussion Point

Hello,
I was wondering if one can give meaning to the Fourier transform of the linear function:

$$\int_{-\infty}^{+\infty} x e^{ikx}\, dx$$

I found that it is $$\frac{\delta(k)}{ik}$$, does this make sense?

vanhees71
Gold Member
2019 Award
This expression doesn't make sense since it's intrinsically undefined. A handwaving way is
$$\int_{\mathbb{R}} \mathrm{d} x x \exp(\mathrm{i} k x)=-\mathrm{i} \frac{\mathrm{d}}{\mathrm{d} k} \int_{\mathbb{R}} \mathrm{d} x \exp(\mathrm{i} k x)=-2 \pi \mathrm{i} \frac{\mathrm{d}}{\mathrm{d} k} \delta(k).$$

• 1 person
This expression doesn't make sense since it's intrinsically undefined. A handwaving way is
$$\int_{\mathbb{R}} \mathrm{d} x x \exp(\mathrm{i} k x)=-\mathrm{i} \frac{\mathrm{d}}{\mathrm{d} k} \int_{\mathbb{R}} \mathrm{d} x \exp(\mathrm{i} k x)=-2 \pi \mathrm{i} \frac{\mathrm{d}}{\mathrm{d} k} \delta(k).$$
Hmm.. seems to make sense. Why is there a minus sign popping up?

mathman
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