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Argument (Complex Number)

A Complex Number $z$ may be represented as

z \equiv x+iy = \vert z\vert e^{i \theta},
\end{displaymath} (1)

where $\vert z\vert$ is called the Modulus of $z$, and $\theta$ is called the argument
\arg(x+iy) \equiv \tan^{-1}\left({y \over x}\right).
\end{displaymath} (2)

$\displaystyle \arg(zw)$ $\textstyle =$ $\displaystyle \arg(\vert z\vert e^{i\theta_z}\vert w\vert e^{i\theta_w}) = \arg(e^{i\theta_z}e^{i\theta_w})$  
  $\textstyle =$ $\displaystyle \arg[e^{i(\theta_z+\theta_w)}] = \arg(z)+\arg(w).$ (3)

Extending this procedure gives
\arg(z^n) = n \arg(z).
\end{displaymath} (4)

The argument of a Complex Number is sometimes called the Phase.

See also Affix, Complex Number, de Moivre's Identity, Euler Formula, Modulus (Complex Number), Phase, Phasor


Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 16, 1972.

© 1996-9 Eric W. Weisstein