1.4: The Principal Argument
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The Argument
In this text the notation arg(z) is used to designate an arbitrary argument of z, which means that arg(z) is a set rather than a number. In particular, the relation
arg(z1)=arg(z2)
is not an equation, but expresses equality of two sets.
As a consequence, two non-zero complex numbers r1(cosφ1+isinφ1) and r2(cosφ2+isinφ2) are equal if and only if
r1=r2, and φ1=φ2+2kπ,
where k∈Z.
In order to make the argument of z a well-defined number, it is sometimes restricted to the interval (−π,π]. This special choice is called the principal value or the main branch of the argument and is written as Arg(z).
Note that there is no general convention about the definition of the principal value, sometimes its values are supposed to be in the interval [0,2π). This ambiguity is a perpetual source of misunderstandings and errors.
The Principal Argument
The principal value Arg(z) of a complex number z=x+iy is normally given by
Θ=arctan(yx),
where y/x is the slope, and arctan converts slope to angle. But this is correct only when x>0, so the quotient is defined and the angle lies between −π/2 and π/2. We need to extend this definition to cases where x is not positive, considering the principal value of the argument separately on the four quadrants.
The function Arg(z):C∖{0}→(−π,π] is defined as follows:
Arg(z)={arctanyxifx>0,y∈Rarctanyx+πifx<0,y≥0arctanyx−πifx<0,y<0π2ifx=0,y>0−π2ifx=0,y<0undefinedifx=0,y=0
Thus, if z=r(cosΘ+isinΘ), with r>0 and −π<Θ<π, then
arg(z)=Arg(z)+2nπ, n∈Z.
We can visualize the multiple-valued nature of by using Riemann surfaces. The following interactive shows some of the infinite values of . Each branch is identified with a different color.