2 definitions for argument, why?

[Jhenrique]

In the wiki, I found this definition for the argument:

http://en.wikipedia.org/wiki/List_of_trigonometric_identities#Exponential_definitions

However, in other page of the wiki (http://en.wikipedia.org/wiki/Complex_conjugate#Use_as_a_variable), I found this definition for argument:$$\arg(z) = \ln(\sqrt[2 i]{z \div \bar{z} }) = \frac{ln(z) - ln(\bar{z})}{2 i}$$I don't understand why exist 2 defitions for the argument and how those 2 defitions are related.

 

[pasmith]

In the wiki, I found this definition for the argument:

This gives the inverse of $\mathrm{cis}\,\theta = \cos \theta + i \sin \theta = e^{i\theta}$. It is not a definition of the argument, but reflects the fact that if $z = e^{i\theta}$ then
$$-i \log e^{i\theta} = -i(i \theta) = \theta = \arg z.$$
It doesn't give $\arg z$ if $|z| = R \neq 1$:
$$-i \log (Re^{i\theta}) = -i \log R + \theta \neq \arg z$$

However, in other page of the wiki (http://en.wikipedia.org/wiki/Complex_conjugate#Use_as_a_variable), I found this definition for argument:$$\arg(z) = \ln(\sqrt[2 i]{z \div \bar{z} }) = \frac{ln(z) - ln(\bar{z})}{2 i}$$I don't understand why exist 2 defitions for the argument and how those 2 defitions are related.

This gives $\arg z$ for any $z \neq 0$ (if you choose the correct branch of $z^{1/(2i)}$).

 

[Jhenrique]

I liked your answer!

 

[Chronos]

There is almost always an alternative way of expressing the same mathematical argument, with a little imagination. It's not always obvious.