Complex Number Argument

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A Complex Number Argument is an angular function and it is obtained from the inverse function of the Tangent Function,

  • AKA: Phase, Arctangent, Phase Angle.
  • Context:
    • It is denoted as [math]\displaystyle{ arg(z) }[/math], where [math]\displaystyle{ z=x+iy }[/math] is a complex number.
    • It can be defined as the counterclockwise angle from the positive real axis to complex number vector in the complex plane
    • It can generally be defined as [math]\displaystyle{ arg (a+ib) =tan^{-1}(x/y)+ 2\pi n }[/math] with [math]\displaystyle{ n = 0, \pm 1, \pm 2 \dots }[/math] and [math]\displaystyle{ tan^{-1}(x/y) }[/math] being the inverse function of the tangent function. When [math]\displaystyle{ \theta=arg(a+ib) }[/math] is such that [math]\displaystyle{ -\pi\lt \theta\lt \pi }[/math], this called the principal value of the argument.
    • It has the following properties:
      • [math]\displaystyle{ arg(zw)=arg(z)arg(w) }[/math] with [math]\displaystyle{ z }[/math] and [math]\displaystyle{ w }[/math] being complex numbers
  • Example(s):
    • [math]\displaystyle{ arg(1)=0 }[/math]
    • [math]\displaystyle{ arg(-1)=\pi }[/math]
    • [math]\displaystyle{ arg(i)=\pi/2 }[/math]
    • [math]\displaystyle{ arg(-i)=-\pi/2 }[/math]
    • [math]\displaystyle{ arg(1+i)=\pi/4 }[/math]
  • Counter-Example(s):
  • See: Complex Number, Complex Exponential Function.

References

2015

  1. Geometrically, in the complex plane, as the angle [math]\displaystyle{ \varphi }[/math] from the positive real axis to the vector representing [math]\displaystyle{ z }[/math]. The numeric value is given by the angle in radians and is positive if measured counterclockwise.
  2. Algebraically, as any real quantity φ such that
[math]\displaystyle{ z = r (\cos \varphi + i \sin \varphi) }[/math]
for some positive real [math]\displaystyle{ r }[/math]. The quantity [math]\displaystyle{ r }[/math] is the modulus of [math]\displaystyle{ z }[/math], denoted [math]\displaystyle{ |z| }[/math]:
[math]\displaystyle{ r = \sqrt{x^2 + y^2}. }[/math]
The names amplitude for the modulus and phase for the argument are sometimes used equivalently.
Under both definitions, it can be seen that the argument of any (non-zero) complex number has many possible values: firstly, as a geometrical angle, it is clear that whole circle rotations do not change the point, so angles differing by an integer multiple of [math]\displaystyle{ 2\pi }[/math] radians (a complete circle) are the same. Similarly, from the periodicity of Sine Function and Cosine Function, the second definition also has this property.
  • (Lee, 2015) ⇒ KK Lee (2015). “STPM Mathematics (T) Term 1 Algebra and Geometry/Chapter 4 Complex Numbers", page 6 [1]
    • QUOTE: Argument of complex number [math]\displaystyle{ z=x+yi }[/math] is the angle [math]\displaystyle{ theta }[/math] between the positivevreal axis and the vectorrepresenting [math]\displaystyle{ z }[/math] on the Argand diagram
For a complex number [math]\displaystyle{ z=x+yi }[/math], [math]\displaystyle{ tan\;\theta=\frac{x}{y} }[/math]
The argument [math]\displaystyle{ \theta }[/math] is such that [math]\displaystyle{ -\pi\lt \theta\lt \pi }[/math] (We call this as principal argument)
The argument of complex number is written as [math]\displaystyle{ arg z }[/math] or [math]\displaystyle{ Arg z }[/math]
The argument of complex number is usually given in radians

2004

[math]\displaystyle{ |z|= \sqrt{x^2 +y^2} }[/math]
The argument of a complex number is the angle which the vector representing that number makes with the positive x-axis.

1999

  • (Wolfram Mathworld, 1999) ⇒ http://mathworld.wolfram.com/ComplexArgument.html : Retrieved:2015-11-19
    • QUOTE: A complex number z may be represented as : [math]\displaystyle{ z=x+iy=|z|e^{i\theta}, }[/math] where [math]\displaystyle{ |z| }[/math] is a positive real number called the complex modulus of [math]\displaystyle{ z }[/math], and [math]\displaystyle{ \theta }[/math] (sometimes also denoted [math]\displaystyle{ \varphi }[/math]) is a real number called the argument. The argument is sometimes also known as the Phase or, more rarely and more confusingly, the amplitude (Derbyshire 2004, pp. 180-181 and 376).

      The complex argument of a number [math]\displaystyle{ z }[/math] is implemented in the Wolfram Language as Arg[z].

      The complex argument can be computed as :[math]\displaystyle{ arg(x+iy)=tan^{-1}(y/x) }[/math]

      Here, [math]\displaystyle{ \theta }[/math], sometimes also denoted [math]\displaystyle{ \varphi }[/math], corresponds to the counterclockwise angle from the positive real axis, i.e., the value of theta such that [math]\displaystyle{ x=cos\theta }[/math] and y=[math]\displaystyle{ sin\theta }[/math].

1996

1991

  • ( Sharma, J. N.1991) ⇒ J.N. Sharma(1991), "Functions of a Complex variable", Krishna Prakashan Media ⇒ http://books.google.ca/books?id=ksafkD8uv1gC
    • QUOTE: … modulus of the complex number [math]\displaystyle{ z }[/math] written [math]\displaystyle{ |z| }[/math] and [math]\displaystyle{ \theta = tan^{-1} y/x }[/math] is called the argument or amplitude of [math]\displaystyle{ z }[/math] written as [math]\displaystyle{ arg\;z }[/math]. It follows that [math]\displaystyle{ |z| = 0 }[/math] if and only if [math]\displaystyle{ x = 0 }[/math] and [math]\displaystyle{ y = 0 }[/math]. Geometrically, [math]\displaystyle{ z }[/math] is the distance of the point [math]\displaystyle{ z }[/math] from the origin. Also argument of a complex number is not unique, since if [math]\displaystyle{ \theta }[/math] be a value of the argument, so also is [math]\displaystyle{ 2n\pi +\theta }[/math] where [math]\displaystyle{ n = 0, \pm 1, \pm 2 \dots }[/math] . The value of argument which satisfies the inequality
[math]\displaystyle{ -\pi\lt \theta\lt \pi }[/math]
is called the principal value of the argument. We remark that argument of 0 is not defined.

1987

[math]\displaystyle{ cos\theta = \frac{x}{\sqrt{x^2+y^2}} \qquad\textrm{and}\qquad sin\theta = \frac{x}{\sqrt{x^2+y^2}}\qquad\textrm{(3)} }[/math]
If [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] are not both zero i.e., if [math]\displaystyle{ z }[/math] is a non-zero complex number, then there exist values of [math]\displaystyle{ \theta }[/math] which satisfy the equations (3) simultaneously. Any value of [math]\displaystyle{ \theta }[/math] satisfying the equations (3) is called an argument or amplitude of the complex number [math]\displaystyle{ z }[/math] and we write
[math]\displaystyle{ \theta = arg z \qquad\textrm{or}\qquad\theta = amp z }[/math]
Argument of a complex number is not unique, since if [math]\displaystyle{ \theta }[/math] be a value of the argument, so also is [math]\displaystyle{ 2\pi n + \theta }[/math], where [math]\displaystyle{ n }[/math] it is any integer.
The value of argument which satisfies the inequality
[math]\displaystyle{ -\pi\lt \theta\lt \pi }[/math]
is called the principal value of the argument. Usually by argument of a complex number we understand its principal value unless stated otherwise.
The zero complex number cannot be put in the form [math]\displaystyle{ r (cos \theta + i sin \theta) }[/math] and thus the argument of zero complex number does not exist i.e., is undefined.

1970

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