Phasor

The representation, beloved of engineers and physicists, of a complex number in terms of a complex exponential

(1)

where i (called j by engineers) is the imaginary number and the complex modulus and complex argument (also called phase) are

			(2)
			(3)

Here, (sometimes also denoted ) is called the complex argument or the phase. It corresponds to the counterclockwise angle from the positive real axis, i.e., the value of such that and . The special kind of inverse tangent used here takes into account the quadrant in which lies and is returned by the FORTRAN command ATAN2(Y,X) and the Wolfram Language function ArcTan[x, y], and is often restricted to the range . In the degenerate case when ,

(4)

It is trivially true that

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Now consider a scalar function . Then

			(6)
			(7)
			(8)
			(9)

where is the complex conjugate. Look at the time averages of each term,

			(10)
			(11)
			(12)
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			(15)
			(16)
			(17)
			(18)

Therefore,

(19)

Consider now two scalar functions

			(20)
			(21)

Then

			(22)
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			(27)

In general,

(28)