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Using complex numbers to represent waves [duplicate]

When talking about a plane wave of the form $$\vec E=\vec E_0\cos(\vec k \cdot \vec r-\omega t)$$ We can replace it by $$\vec E=\vec E_0\exp[i(\vec k \cdot \vec r-\omega t)]$$ so that it is easier for ...
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Using complex exponential to represent waves in EM [duplicate]

Ever since we've been using exponentials to work with electromagnetic waves, I've been confused about the imaginary portion and want to confirm my thinking. What does the imaginary portion represent? ...