# Talk:Argument (complex analysis)

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Field: Basics

## Accuracy (?? Anon)

One should probably add a remark concerning the difference between Arg(z) and arg(z) (one is a set and one is an element of the reals).

$\operatorname{arg}(x+\mathrm{i}y) = 2\arctan \frac{y}{\sqrt{x^2 + y^2} + x}$

It's a closed expression for the whole domain of the function except y=0 && x$\leq$0. The page about atan2 uses that expression. Geek1337 (talk) 18:38, 24 September 2008 (UTC)

\begin{align} \text{Let}\ z & = x + \mathrm{i}y \\ |z| & = \exp[\Re(\ln z)] \\ \operatorname{arg}(z) & = \Im(\ln z)\,\bmod\,\pi \end{align}
Cheers, The Doctahedron, 23:35, 12 January 2012 (UTC)

## programing language

Hi. Here is the code in Maxima :

/* principial value of argument of complex number in turns  */
carg_t(z):=
block(
[t],
t:carg(z)/(2*%pi),  /* now in turns */
if t<0 then t:t+1, /* map from [-1/2,1/2) to [0, 1) */
return(t)
)\$


Can I put it on page ?--Adam majewski (talk) 14:11, 11 January 2009 (UTC)

This article is about the mathematical function. The atan2 function is the one used normally when computing arg. I can't see the point in putting code to derive it there either since the definition is perfectly good enough for a person to do that, and anyway most systems provide an atan2 to do the job faster and better. Plus the definition of arg and atan2 is normally from -pi to pi and you have explicitly changed that to something different. Plus Maxima isn't exactly mainstream. Dmcq (talk) 15:14, 11 January 2009 (UTC)
Diagram of Atan2 made using Maxima and Gnuplot
• carg is function giving argument of complex number. it probably uses atan2.
• I have changed bad description in code . Now it shows that it uses turns as a units of arg. It is common practice in dynamical systems, complex analysis
• Maxima is a free, open source CAS. It uses Lisp. Do you know better programs with that properities ?

--Adam majewski (talk) 16:40, 11 January 2009 (UTC)

carg obviously stands for complex argument and gives the correct result by the definition. At best one could say is that carg implements arg which isn't exactly saying much. Turns is not the result defined for the result of arg whatever about its uses. I really can't see what your point is or why you would want to stick this in. Dmcq (talk) 18:02, 11 January 2009 (UTC)
See Comparison of computer algebra systems for various different CAS including open ones. Dmcq (talk) 18:08, 11 January 2009 (UTC)
OK. Thx.--Adam majewski (talk) 15:45, 15 January 2009 (UTC)

## Rewrite (2009)

The article there was a bit piecemeal and generally needed quite bit of work to firstly make it accurate, and secondly, it still wasn't very helpful to someone new to the subject. I have done essentially a total rewrite working from what was there (the good stuff—quite a bit of it was inaccurate or even wrong). Please do comment here or on my talk, as I do want feedback after doing something like that.— Kan8eDie (talk) 02:11, 11 February 2009 (UTC)

Can't say I think much of the rewrite really.
You have removed the range (-π,π] and replace it with an open interval as far as I can see.
The π++ is new to me so it can't be at all common. I'd distrust a source like that as a general reference.
Azimuth and anomaly are astronomical terms for angles and not functions of a complex number. Amplitude was an old variant of argument, the Knopp book is quite a bit older than the 1996 of Dover's reprint. Phase is also an acceptable alternate more often used in electronics where the ofen se complex numbers.
The period of tan is π not 2π. Putting in that tan(y/x) just causes trouble and is why atan2 was defined..
References need to be a bit better than referring to mathematics dictionary (2002). There's loads of mathematics dictionaries.
You've stuck in mathematics symbolism in the definition that I feel is unwarranted and unnecessary at the level of a reader that would want to look this up. A prime aim of wikipedia is to be accessible. Dmcq (talk) 03:53, 11 February 2009 (UTC)
Thanks for the comments. I'll see what I can do.
Open interval: Yes, we want a holomorphic branch. Adding in π loses continuity. Spot the extension in section 3.
Typo; thanks. The dictionary is a good one. I have found out why I put it there: when I tap AltGr to switch to greek keyboard, it remaps the keys; the XKBD polytonic layout puts + on the } key (Dvorak), so it was not wonder that I tapped that key twice after the π!
Yes, they are old terms, but attested:

amplitude, n. ... 2 the angle between the positive real axis and the vector representing a given complex number in the Argand plane [etc.] also called argument, anomaly, or azimuth.

Interestingly, the dictionary (1st ed 1989) has 'argument, see AMPLITUDE', not the other way around.
Regarding tan, could have been clearer, but it certainly is 2π-periodic (tan x = tan(x + 2π)). The definition omitted to exclude the third quadrant though, which was extremely sloppy, and is now fixed.
Refs are weak. I'll pop into the library and get a couple of undergrad analysis books and add the details in. A dictionary gives the facts, and the rest follows pretty quickly, but I shall add some more stuff in if you think it needs it.
For the 'unnecessary detail', could you give more specific advice? I have included captioned diagrams and explanation to try to make it clear what exactly is going on, which was not in the old article. I have also taken care to introduce technical stuff with phrases like 'formally...' so that the uninterested reader can skip that paragraph. Basically, Riemann surfaces free you from a global topology embeddable in R^3, so there comes a point when thinking in terms of the Argand plane is just not right; that is not where arg really 'lives'. Giving only the first definition is just somehow not right, as it only conveys half the structure of the manifold. This might not make sense to most readers, but maths doesn't work without definitions, and we can box them up, separate them, put them wherever you think they should go (certainly out of the lead—I used friendly text!), but they need to be in the article somewhere.
If I have used any confusing or difficult notation, please do change it. I remembered not to use \forall and \exists and most things that could be in symbols are typed out in full. I am keen to make it readable.— Kan8eDie (talk) 07:33, 11 February 2009 (UTC)
I'll have a go at it after you've finished. however as to your points.
Azimuth and anomaly are not equivalent to argument. They are normally measured in degrees, they don't normally refer to complex numbers and azimuth in particular tend to be measured in a clockwise direction from north. Amplitude and phase have been used as a function and in ways equivalent to argument.
The period of tan is π not 2π and saying it is 2π is to cause trouble.
tan goes wrong when x is negative no just in the third quadrant.
arg is not holomorphic nor does it form a Riemann surface. A requirement for those is that the target be the complex numbers.
I don't know where you got this requirement that the function only be defined in a domain where it is continuous so the negative axis is removed.
You can look up a lot of the stuff and get some references using google scholar or books if you don't have any of your own books. This might help rather than trying to learn everything from a dictionary.
Dmcq (talk) 19:36, 11 February 2009 (UTC)
Well, thanks for that. Some points unavoidable; the rest will teach me to tread more carefully round next year's courses when I haven't read the books yet! I'm not surprised, on reflection, that I made a mess of myself there. My supervisors will happily attest that I sometimes write gibberish even on the stuff I have understood.
I'll let you correct the dictionary on that one. They are directly defined as I gave, but I must say that I have never come across anomaly, and azimuth only in the context of spheres.
I appreciate that. I will try and reword it to make it clearer. I was trying to say that 2π is a period, not that it is the period (the smallest non-negative one).
I fixed up the definition but not the text. Good spot. To be completely thorough, it is not just negative x, but for y too!
Actually, I think Beardon was keen on that. (I will need to check though.)
People don't buy books when there is a library on every street corner, but that does mean that they aren't exactly by my bedside.
In any case, I still think the article is improved from where it was, but it has been a less than sterling effort on my part. Please excuse the unmitigatedly village night that started this all off.— Kan8eDie (talk) 23:15, 11 February 2009 (UTC)
Another thing probably just a slip-up you have the rin the second part of the definition you put in on the side where it would be the inverse of the radius. That's going to be confusing. The third part of the definition doesn't distinguish between positive and negative y. Did you really get this definition out of a dictionary? It is really quite a bad one. Dmcq (talk) 23:39, 11 February 2009 (UTC)
In fact if you just had the second part and removed the other two and put the r on the other side it would be shorter and much better. Dmcq (talk) 23:42, 11 February 2009 (UTC)
Of course you are right. Crazy as it sounds, I do have the dictionary open in front of me, and it gives the full set of conditions (minus the typo). Clearly the quality of the other place is showing itself up (both authors are at Oxford).— Kan8eDie (talk) 22:08, 13 February 2009 (UTC)

Just noticed this discussion now. I left a note on Wikipedia:Requests_for_feedback#arg_.28mathematics.29. bamse (talk) 11:25, 12 February 2009 (UTC)

## Arg/arg

As far as I understand, for the purpose of this article small "arg" is the mulitvalued function and capital Arg the well-defined function. To be consistent we should change all the "arg" into "Arg" in the "Identities" section then, I believe. bamse (talk) 13:44, 15 February 2009 (UTC)

Wikipedia isn't a textbook, it should reflect general practice rather than being consistent. arg is generally used for both except if you're being formal and have to distinguish between them. Dmcq (talk) 23:02, 15 February 2009 (UTC)
Alright, after all it states at the beginning of that section that it is the principal value arg. One more thing concerning amplitude. How about putting a warning as in Amplitude: In older texts the phase is sometimes very confusingly called the amplitude.? In fact, I had never seen the phase being called amplitude before. Being a physicist it is very confusing to me. Maybe mathematicians call it this way!? bamse (talk) 09:32, 16 February 2009 (UTC)
Seems reasonable, it is confusing. It definitely has been called that and not all that uncommonly in the past,I wonder why anyone ever called it that never mind calling it argument. Some of the words in mathematics are very strange. Dmcq (talk) 13:03, 16 February 2009 (UTC)
In fact when talking about Jacobi elliptic functions there is a corresponding angle which is still talked of as the amplitude. Dmcq (talk) 13:13, 16 February 2009 (UTC)

## Definition

The definition is not correct, for instance:

$\!z1=1+i, \arctan(1/1) = \arg z1 = \pi/4$
$z2=-z1, \arctan (-1/-1) = \pi/4 \neq \arg z2$

The definition has to distinguish several possibilities. Nijdam (talk) 21:55, 5 October 2011 (UTC)

You're quite right, the article was changed by someone in the last month to stick in that arctan. atan2 would do the trick but some people have this idea that atan2 isn't mathematical and insist on using arctan with the understanding that they just choose the angle in the way that atan2 does. It is really silly. I hate to think how many computer programs for instance have gone wrong because of people using arctan instead of atan2 as so many textbooks do this silly thing. I'll go and remove that arctan. Dmcq (talk) 00:17, 6 October 2011 (UTC)