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Complex argument (arg)

The complex argument of a complex number gives the angle θ with the x-axis



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



It applies θ = arccis x = arg x


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