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 Modulus and Argument (also called Phase) are
Here, is the counterclockwise Angle from the Positive Real axis. In the degenerate
case when ,
|
(4) |
It is trivially true that
|
(5) |
Now consider a Scalar Function
. Then
Look at the time averages of each term,
|
(7) |
|
(8) |
|
(9) |
Therefore,
|
(10) |
Consider now two scalar functions
Then
In general,
|
(15) |
See also Affix, Argument (Complex Number), Complex Multiplication, Complex Number, Modulus
(Complex Number), Phase
© 1996-9 Eric W. Weisstein
1999-05-26