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In this pdf https://studylib.net/doc/18704299/chapter-1 the author writes that the Fourier transform of the current density is: $$ \Delta j_{i}(\mathbf{k}, \omega) \equiv \frac{1}{(2 \pi)^{2}} \iiiint e^{-i(\mathbf{k} \cdot \mathbf{r}-\omega t)} \Delta j_{i}(\mathbf{r}, t) d^{3} r d t $$ which looks like an ordinary Fourier transform. However, in the same calculations, they write that the Fourier transform of the electrical conductivity function is: $$ \widetilde{\sigma_{i j}}(\mathbf{k}, \omega) \equiv \iiiint \sigma_{i j}\left(\mathbf{r}^{\prime \prime}, t^{\prime \prime}\right) e^{-i\left(\mathbf{k} \cdot \mathbf{r}^{\prime \prime}-\omega t^{\prime \prime}\right)}(-1)^{4} d^{3} r^{\prime \prime} d t^{\prime \prime} $$ It thus seems that the Fourier transform constant is here \$(-1)\$ rather than \$\frac{1}{\sqrt{2}}\$, which doesn't seem to make sense since both Fourier transforms are used in the same derivation. The end result is correct, but I don't understand how different conventions of the Fourier transform can be mixed in the same calculations?

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    \$\begingroup\$ On EE.SE use \$ for inline MathJAX. Welcome to EE.SE. \$\endgroup\$
    – Transistor
    Mar 5 at 17:29

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