Editing Electrical impedance (section)

=== Validity of complex representation ===
This representation using complex exponentials may be justified by noting that (by [[Euler's formula]]):

:<math>\ \cos(\omega t + \phi) = \frac{1}{2} \Big[ e^{j(\omega t + \phi)} + e^{-j(\omega t + \phi)}\Big]</math>

The real-valued sinusoidal function representing either voltage or current may be broken into two complex-valued functions. By the principle of [[superposition principle|superposition]], we may analyse the behaviour of the sinusoid on the left-hand side by analysing the behaviour of the two complex terms on the right-hand side. Given the symmetry, we only need to perform the analysis for one right-hand term. The results are identical for the other. At the end of any calculation, we may return to real-valued sinusoids by further noting that

:<math>\ \cos(\omega t + \phi) = \operatorname{Re} \Big\{ e^{j(\omega t + \phi)} \Big\}</math>