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

(5)

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)
			(13)
			(14)
			(15)
			(16)
			(17)
			(18)

Therefore,

(19)

Consider now two scalar functions

			(20)
			(21)

Then

			(22)
			(23)
			(24)
			(25)
			(26)
			(27)

In general,

(28)

REFERENCES:

Krantz, S. G. "Polar Form of a Complex Number." §1.2.4 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 8-10, 1999.