2 definitions for argument, why?


by Jhenrique
Tags: argument, definitions
Jhenrique
Jhenrique is offline
#1
Feb17-14, 11:18 AM
P: 427
In the wiki, I found this definition for the argument:



http://en.wikipedia.org/wiki/List_of...al_definitions

However, in other page of the wiki (http://en.wikipedia.org/wiki/Complex..._as_a_variable), I found this definition for argument:[tex]\arg(z) = \ln(\sqrt[2 i]{z \div \bar{z} }) = \frac{ln(z) - ln(\bar{z})}{2 i}[/tex]I don't understand why exist 2 defitions for the argument and how those 2 defitions are related.
Phys.Org News Partner Mathematics news on Phys.org
Hyperbolic homogeneous polynomials, oh my!
Researchers help Boston Marathon organizers plan for 2014 race
'Math detective' analyzes odds for suspicious lottery wins
pasmith
pasmith is offline
#2
Feb17-14, 11:57 AM
HW Helper
P: 775
Quote Quote by Jhenrique View Post
In the wiki, I found this definition for the argument:

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

However, in other page of the wiki (http://en.wikipedia.org/wiki/Complex..._as_a_variable), I found this definition for argument:[tex]\arg(z) = \ln(\sqrt[2 i]{z \div \bar{z} }) = \frac{ln(z) - ln(\bar{z})}{2 i}[/tex]I don't understand why exist 2 defitions for the argument and how those 2 defitions are related.
This gives [itex]\arg z[/itex] for any [itex]z \neq 0[/itex] (if you choose the correct branch of [itex]z^{1/(2i)}[/itex]).
Jhenrique
Jhenrique is offline
#3
Feb17-14, 06:45 PM
P: 427
I liked your answer!

Chronos
Chronos is offline
#4
Feb17-14, 08:54 PM
Sci Advisor
PF Gold
Chronos's Avatar
P: 9,183

2 definitions for argument, why?


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


Register to reply

Related Discussions
Analytical definitions vs intuitive (or perhaps "first year") definitions Calculus 1
2 definitions Calculus & Beyond Homework 1
I used the definitions, now what? Calculus & Beyond Homework 3
A few definitions Cosmology 0
Looking for definitions Advanced Physics Homework 0