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Complex argument (arg)
The complex argument of a complex number gives the angle θ with the x-axis
Explanation
The complex number z is written with fasor notation as
where |z| is a positive real number, and you call this the modulus of z. The Greek letter θ (theta) is a real number that specifies the angle with the x-axis, and is called the argument. Thus x = |z| cos θ and y = |z| sin θ.
The argument of z in the interval [0, 2π) is called the principle value. As principle value also the interval (−π, π] can be selected. For a point on the y-axis, where x = 0, holds
The complex argument is calculated as
Special values of the complex argument are
From the definition follows that the product of two complex numbers (z ≠ 0) is equal to the sum of their arguments
It follows
with as special case
A division of two complex numbers gives
Example 1
Example 2
Using arg (z) and modulus |z| the complex number z can be written as
Example 3
If z is not a pure imaginary number, so is not on the vertical y-axis, it holds
Details
It applies θ = arccis x = arg x