Maths Spl

A nonzero complex numbers has infinitely many arguments. They all differ by integer multiples of 2. If you think of the argument geometrically, they are all co-terminal angles.
The principal argument is unique and is most commonly taken to be that argument in the interval (-,]. You should check with a particular author to see what interval is being taken, but (-, ] is pretty standard.
As an example, consider the number 1 + i. This number plotted in the plane would live on the line y = x at a distance (2) from the origin. So, a polar representation for this number is
1 + i = (2) e^(i/4)
Of course e^(i) is 2-periodic, so you can write
1 + i = (2) e^(i (/4 + 2k)) where k is any integer.
The argument (lower case "a") is multivalued
arg (1 + i) = /4 + 2k.
The principal argument (upper case "A") is the one number of the form /4 + 2k that satisfies - Arg(1 + i). In particular,
Arg(1 + i) = /4.