Another Formula for an Argument

In a previous article, I showed the error in the ridiculous formula for the argument of a complex number that you can find on various websites:

in the article I found a more accurate formula (although certainly more complicated):

(read my article "A Formula for an Argument" to find out why this works)

However, although this formula is correct, there is a better way to go about producing a formula for the argument of a complex number, (by ‘better’ I merely mean a way that gives a simpler result).

The first (incorrect) formula above has the problem that it is only valid for half of the possible values of an argument of a complex number (only between minus a half of Pi to a half of Pi). It completely ignores the rest!!