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.

 

[CellCoree]

I can explain that the two definitions for argument are related to different mathematical concepts and are used for different purposes. The first definition, found in the list of trigonometric identities, is related to the concept of complex numbers and their representation in polar form. In this context, the argument represents the angle between the complex number and the positive real axis.

The second definition, found in the page on complex conjugates, is related to the concept of logarithms and their use in solving equations involving complex numbers. In this context, the argument represents the phase angle of the complex number and is used to find the complex conjugate of a number.

It is important to note that both definitions are valid and useful in their respective contexts. The first definition is commonly used in trigonometry and calculus, while the second definition is commonly used in complex analysis and engineering.

Overall, the existence of two definitions for the argument highlights the versatility and applicability of complex numbers in various fields of mathematics and science. It is not uncommon for different definitions or representations of a concept to coexist and be used in different contexts.