Editing Complex number (section)

====Electromagnetism and electrical engineering====
{{Main|Alternating current}}
In [[electrical engineering]], the [[Fourier transform]] is used to analyze varying [[electric current]]s and [[voltage]]s. The treatment of [[resistor]]s, [[capacitor]]s, and [[inductor]]s can then be unified by introducing imaginary, frequency-dependent resistances for the latter two and combining all three in a single complex number called the [[Electrical impedance|impedance]]. This approach is called [[phasor]] calculus.

In electrical engineering, the imaginary unit is denoted by {{mvar|j}}, to avoid confusion with {{mvar|I}}, which is generally in use to denote electric current, or, more particularly, {{mvar|i}}, which is generally in use to denote instantaneous electric current.

Because the voltage in an AC circuit is oscillating, it can be represented as

<math display=block> V(t) = V_0 e^{j \omega t} = V_0 \left (\cos\omega t + j \sin\omega t \right ),</math>

To obtain the measurable quantity, the real part is taken:

<math display=block> v(t) = \operatorname{Re}(V) = \operatorname{Re}\left [ V_0 e^{j \omega t} \right ] = V_0 \cos \omega t.</math>

The complex-valued signal {{math|''V''(''t'')}} is called the [[analytic signal|analytic]] representation of the real-valued, measurable signal {{math|''v''(''t'')}}.
<ref>{{cite book |last1=Grant |first1=I.S. |title=Electromagnetism |year=2008|edition=2 |publisher=Manchester Physics Series |isbn=978-0-471-92712-9 |last2=Phillips |first2=W.R.}}</ref>