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my question is very simple. Physicists use this notation in order to write a (for example) plane wave:

$$ \xi(z) = \xi^+ \mathrm{e}^{+\mathrm{i}kz} + \xi^- \mathrm{e}^{-\mathrm{i}kz}, $$ where $\xi^+$ is the progressive wave and $\xi^-$ is the regressive wave.

Engineers adopt a different notation:

$$ \xi(z) = \xi^+ \mathrm{e}^{-\mathrm{i}kz} + \xi^- e^{+\mathrm{i}kz}. $$

Basically you can move from one notation to the other by choosing $\mathrm{i} \to -\mathrm{i}$, or vice versa.

Obviously both are correct, but I find a little more sense in the engineers' notation, because $\xi(z)$ can be interpreted as a phasor, and in order to come back in the time domain you have to multiply for $\mathrm{e}^{\mathrm{i}\omega t}$, and this is the same thing that you do when you do the inverse Fourier transform, in order to get a function in the time domain:

$$ f(t) = \frac{1}{\sqrt{2 \pi}}\int_{-\infty}^{+\infty} F(\omega) \ \mathrm{e}^{\mathrm{i} \omega t} \mathrm{d}\omega, $$ while the physicists have to multiply for $\mathrm{e}^{-\mathrm{i}\omega t}$, in order to have $\xi^+$ as the coefficient of the progressive wave and $\xi^-$ as the coefficient of the regressive wave.

Obviously this is not fundamental, because the only thing important is to have a $f(kz-\omega t)$ as the progressive wave and $f(kz+\omega t)$ as the regressive wave, but I find more natural the engineers' notation. Is there a reason why physicists use another notation?

A friend of mine told me that in origin engineer chose another imaginary root, but I'm not convinced. He basically told me that since the equation: $$ x^2 = -1 \Longrightarrow \left| x \right| = \sqrt{-1} $$ has two roots, namely: $x_1 = \sqrt{-1}$ and $x_2 = -\sqrt{-1}$, physicists and mathematicians chose $\mathrm{i} = \sqrt{-1}$, while engineers chose $j = -\sqrt{-1}$.

The reason why I'm not convinced is that they write other quantities in the same way, for example you can find $Z_L = \mathrm{i}\omega L$ and $Z_L = j\omega L$ in physics and engineering books.

Do you have any idea why this discrepancy does exists? Can you cite some source about the notation discrepancy? I didn't find anything.

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    $\begingroup$ Because physicists and engineers have different goals and applications, and hence have to deal with different uses of the formulas more often. For more theoretical purposes it is cleaner to have the indices match the signs in the exponents, the need to apply inverse transform to actually recover the function is less pressing in physics. Btw, $i$ and $j$ both denote $\sqrt{-1}$, electrical engineers tend to use $j$ simply because $i$ is reserved for current, and they often have to deal with formulas that have both. Notational variations for practical reasons are very common. $\endgroup$ – Conifold Dec 22 '20 at 22:09
  • $\begingroup$ Also see my previous answer regarding $i$ and $j$: hsm.stackexchange.com/questions/9654/… $\endgroup$ – M. Farooq Dec 23 '20 at 7:06
  • $\begingroup$ The first rule of natural sciences: if there is a sign in an equation, there are at least different versions of the equation exist in two different fields, depending what they felt natural. $\endgroup$ – Greg Dec 31 '20 at 12:37

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