Complex Number Argument

The argument is an angle \( \theta \) qualifying the complex number \( z \):

$$ \arg(z) = 2\arctan \left(\frac{\Im(z)}{\Re(z) + |z|} \right) = \theta \mod 2\pi $$

with \( \Re(z) \) the real part and \( \Im(z) \) the imaginary part of \( z \).

Example: Take \( z = 1+i \), the real part is \( 1 \), the imaginary part is \( 1 \) and the modulus of the complex number \( |z| \) equals \( \sqrt(2) \), so \( \arg(z) = 2 \arctan \left( \frac{1}{1 + \sqrt(2) } \right) = \frac{\pi}{4} \)

The result of the \( \arg(z) \) calculation is a value between \( -\pi \) and \( +\pi \) and the theta value is modulo \( 2\pi \)

The argument of \( 0 \) is \( 0 \).

Take \( z \), \( z_1 \) and \( z_2 \) be non-zero complex numbers and \( n \) is a natural integer. The remarkable properties of the argument function are:

$$ \arg( z_1 \times z_2 ) \equiv \arg(z_1) + \arg(z_2) \mod 2\pi $$

$$ \arg( z^n ) \equiv n \times \arg(z) \mod 2\pi $$

$$ \arg( \frac{1}{z} ) \equiv -\arg(z) \mod 2\pi $$

$$ \arg( \frac{z_1}{z_2} ) \equiv \arg(z_1) - \arg(z_2) \mod 2\pi $$

If \( a \) is a strictly positive real and \( b \) a strictly negative real, then

$$ \arg(a \cdot z) \equiv \arg(z) \mod 2\pi $$

$$ \arg(b \cdot z) \equiv \arg(z) +\pi \mod 2\pi $$