= Fourier mode
{c}
{title2=$e^{i\omega x}$}
A <Fourier mode> is a single complex oscillatory component $e^{i\omega x}$. Its real and imaginary parts are a <cosine> and a <sine>; $\omega$ is the spatial <angular frequency>. Every constant-coefficient <differential operator> acts on it by multiplication: $P(\partial_x)e^{i\omega x}=P(i\omega)e^{i\omega x}$. In particular, $\partial_x^2e^{i\omega x}=-\omega^2e^{i\omega x}$. These <eigenfunctions> reduce a <partial differential equation> such as the <heat equation> to separate equations for each <frequency>. On the real line a pure <Fourier mode> is not in the <L2 space>; the <Fourier transform> uses such modes in a continuous superposition.
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