The representation, beloved of engineers and physicists, of a complex number in terms of a complex exponential
(1)
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where i (called j by engineers) is the imaginary number and the complex modulus and complex argument (also called phase) are
(2)
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(3)
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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)
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It is trivially true that
(5)
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Now consider a scalar function . Then
(6)
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(7)
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(8)
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(9)
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where is the complex conjugate. Look at the time averages of each term,
(10)
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(11)
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(12)
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(13)
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(14)
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(15)
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(16)
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(17)
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(18)
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Therefore,
(19)
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Consider now two scalar functions
(20)
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(21)
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Then
(22)
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(23)
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(24)
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(25)
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(26)
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(27)
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In general,
(28)
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