Complex Modulus
المؤلف:
Abramowitz, M. and Stegun, I. A.
المصدر:
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover
الجزء والصفحة:
...
18-10-2018
1003
Complex Modulus
The modulus of a complex number 
, also called the complex norm, is denoted 
and defined by
 |
(1)
|
If 
is expressed as a complex exponential (i.e., a phasor), then
 |
(2)
|
The complex modulus is implemented in the Wolfram Language as Abs[z], or as Norm[z].
The square 
of 
is sometimes called the absolute square.
Let 
and 
be two complex numbers. Then
so
 |
(5)
|
Also,
so
 |
(8)
|
and, by extension,
 |
(9)
|
The only functions satisfying identities of the form
 |
(10)
|
are 
, 
, and 
(Robinson 1957).
REFERENCES:
Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 16, 1972.
Krantz, S. G. "Modulus of a Complex Number." §1.1.4 n Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 2-3, 1999.
Robinson, R. M. "A Curious Mathematical Identity." Amer. Math. Monthly 64, 83-85, 1957.
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