In physics, when we solve an PDE or ODE, the solution usually has the form of
\begin{equation}
f=C_+e^{i\lambda x}+C_-e^{-i\lambda x}
\end{equation}
and the "causility" will eliminate one term as it violates the "physics".
I am wondering how the "causility" is defined here. In detail, I will focus on the time harmonic term as it directly reflects the "causility".
As for time harmonic, the harmonic term usually takes the form
\begin{equation}
T(t)\sim e^{-i\omega t}
\end{equation}
However, in some books or papers, there seems to be another set of notations, they replace the imaginary unit $i$ with $-j$, which reads
\begin{equation}
T(t)\sim e^{j\omega t}
\end{equation}
I've heard some explanations on this topic like:
- Both $i$ and $j$ are square roots of $-1$, but "$i$" is "$+\sqrt{-1}$" while "$j$" stands for "$-\sqrt{-1}$".
- In engineering, especially electronic engineering, "$i(t)$" is preseved as the transient current. So those people will use "$j$" as the imaginary unit.
Both notion sets actually work for me. However, during review tasks, I saw a few (very few) papers were parepared with notation \begin{equation} T(t)\sim e^{i\omega t} \end{equation} With this notion set, the harmonic phase development will be quite confusing, the phase term can be any combination of: \begin{equation} \pm kr \pm \omega t \end{equation}
Here, my question is:
Why is time harmonic term $e^{-i\omega t}$ rather than $e^{i\omega t}$, is there any reason for this or is it idiomatic since the first guy (who's that guy?)