Argument (complex Analysis) - Identities

Identities

One of the main motivations for defining the principal value Arg is to be able to write complex numbers in modulus-argument form (the modulus of z = x + iy is |z| = √(x2 + y2), the length of the vector on the Argand diagram). Hence for any complex number z,

.

This is only really valid if z is non-zero but can be considered as valid also for z = 0 if Arg(0) is considered as being an indeterminate form rather than as being undefined.

Some further identities follow. If z₁ and z₂ are two non-zero complex numbers then

If z ≠ 0 and n is any integer then