Arg (mathematics)

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In mathematics the arg function is a logical function that extracts the angular component of a complex number or function.[1] The angular component is also referred to as the argument. For real numbers x and y, arg(x + iy) is equivalent to the function atan2(y, x), which is constrained to the range (−π, π]. That is, for y ≠ 0:

\arg(x+iy) = \begin{cases} \phi\cdot \sgn(y) & \qquad x > 0 \\ \frac{\pi}{2}\cdot \sgn(y) & \qquad x = 0 \\ (\pi - \phi)\cdot \sgn(y) & \qquad x < 0 \\ \end{cases}

where \phi\, is the angle in [0,π/2) such that:  \tan(\phi) = \left| \frac{y}{x}\right|.\,  And sgn is the sign function.

And:

\arg(x+i0) = \begin{cases} 0 & \qquad x > 0 \\ \text{undefined} & \qquad x = 0 \\ \pi & \qquad x < 0 \\ \end{cases}

This produces results in the range (−π, π], which can be mapped to [0, 2π) by adding 2π to the negative values.

arg is also used less formally to represent an unconstrained angle. For instance, when:

  • \phi(t)\, is a continuous function of time (such as \omega t)\,,
  • and z(t) = r\,\mathrm{e}^{i \phi(t)}\,  (called exponential form),
  • or z(t) = r\,(\cos \phi(t) + i\sin \phi(t))\,  (called trigonometric form),

arg(z(t)) often denotes the continuous function, \phi(t).\,

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[edit] Alternative implementation

If r = \sqrt{x^2+y^2} is readily available, a potentially simpler implementation of arg(x + iy) is also available.

For y ≠ 0:

\arg(x + iy) = \theta \cdot \sgn(y),\,

where \theta\, is the angle in [0,π) such that:  \cos(\theta) = \frac{x}{r}.\,

And  \arg(x+i0)\,  is defined as before.

[edit] arg(0 + i 0)

When x and y are both zero,  r = 0,\,  and any angle \phi\, satisfies:

x+iy = r\,\mathrm{e}^{i \phi}.\, 

 

 (Eq.1)

 

Therefore, arg(0 + i0) is sometimes defined as 0, for the sake of uniqueness. However, solving Eq.1 for \phi\, gives:

\arg(x + iy) = \phi = -i\log_e\frac{x+iy}{r}, \,

which is indeterminate/undefined when r=0. In this viewpoint, arg(x + iy) is not necessarily perceived as an angle.[1]

[edit] Notes

  1. ^ a b An alternative definition is available at Complex argument (continued fraction). The essential difference is the treatment of arg(0 + i0).

[edit] External links