# Time-harmonic Maxwell's equations

My question is about the time-harmonic or frequency domain (differential) form of Maxwell's equations. If one solves these and obtain the complex vector field $E$, are the real and imaginary parts independent or correlated? In other words, knowing the real part of the solution, does this put constraints for the imaginary part and vice versa?

• If you assume that the field is truly time harmonic complex then its real and imaginary parts are independent of each other. – hyportnex Jun 30 '15 at 16:18
• If your signal (field), say $E(t)=a(t)cos(\omega t+\phi(t))$ has $s(t) = a(t)cos(\phi (t)$ and $r(t)= a(t)sin(\phi(t)$ such that their bandwidth is much less than $\omega$ then you can say that for the complex envelope $e(t) = s(t)+ \iota r(t)$ of $E(t)$ the real and imaginary parts $s(t)$ and $r(t)$ are Hilbert transforms of each other. – hyportnex Jun 30 '15 at 16:29
• @user31748 Is there a proof for your first comment in the case of a purely time-harmonic field? – Jim Jun 30 '15 at 19:49

No, because the coefficients of any Fourier expansion are usually independent of one other (unless some further conditions hold). The decomposition into real and imaginary part (or equivalently sine and cosine) is a special case of a more general Fourier decomposition $$\psi(\textbf{x},t)=\int d^3k\,d\omega\,\tilde{c}(\textbf{k},\omega)\,\textrm{e}^{-i(\textbf{k}\cdot\textbf{x} - \omega t)}$$ where the exponentials $\textrm{e}^{-i(\textbf{k}\cdot\textbf{x} - \omega t)}$ form a basis of a $L^1(\mathbb{R}^3)$ Hilbert space and are therefore independent of one other (otherwise they would not be a basis).