Talk:Arctangent Logarithmic Formulation

Good as a first stab, but notice needs to be made of the fact that arctan is a multifunction. I call attention to Definition:Arctangent#Caution, for instance. Similar applies to other inverse trig functions. --prime mover (talk) 06:57, 14 December 2012 (UTC)

The logarithm is also a multifunction; as it happens, the identity holds as an equality of multifunctions. I have the impression though that insufficient caution with the peculiarities of complex logarithm is taken... It really cannot be well-defined on all of $\C$. --Lord_Farin (talk) 12:01, 14 December 2012 (UTC)

Are we using arctan to mean inverse tan now? --GFauxPas (talk) 14:33, 14 December 2012 (UTC)

We ought to. $\tan^{-1}$, as has been pointed out on the Tangent page, is misleading notation. arctan is de rigueur. --prime mover (talk) 14:42, 14 December 2012 (UTC)

Having said that, $\tan^{-1} (x)$ is a set: $\{y \in \C: \tan y = x\}$ which is well-defined. It's more complicated than it looks on the surface, and is tractable, but requires considerable fundamental underpinning. --prime mover (talk) 14:47, 14 December 2012 (UTC)

If we're using $y = \tan^{-1} x$ to mean $(x,y) \in \tan$, I object to using "$\iff$". (Actually, I object to even using $"="$, but that's a losing battle) --GFauxPas (talk) 14:53, 14 December 2012 (UTC)

I changed it to $ \arctan x $ (and the others too). — Timwi (talk) 19:33, 14 December 2012 (UTC)

I'm actually unhappy about the entire page until we have covered the multifunctional nature of logarithm in the complex plane. These grade-school definitions are all very well but they are a bit limiting and need to be expanded. This is one reason why this area has not been done; I was nerving myself up to posting up the fundamentals but got bogged down in the topology which I failed at badly. --prime mover (talk) 19:42, 14 December 2012 (UTC)

Well, since this is a wiki, you are free to edit and improve the pages. However, for this one, I’m not sure I see why you need a multifunctional logarithm. This theorem and its proof assume a single-valued logarithm of a complex number, namely:

$\ln(a+bi) = \ln\left({\sqrt{a^2 + b^2} }\right) + i \begin{cases} \frac \pi 2 & : & a = 0 \\ \arctan\left({\frac b a}\right) & : & \mbox{otherwise} \end{cases} $

— Timwi (talk) 11:32, 15 December 2012 (UTC)

My books must all be wrong then. Sorry. I will amend them all to stub out all that rubbish that $\ln$ is a multifunction, e.g. "If $e^z = w$, then we write $w = \ln z$, called the natural logarithm of $z$. Thus the natural logarithmic function is the inverse of the exponential function and can be defined by

$w = \ln z = \ln r + i(\theta+2k\pi)\quad k=0,\pm1,\pm2,\ldots$

where $z = re^{i\theta}=re^{i\theta+2k\pi}$. Note that $l\ln z$ is a multiple-valued (in this case infinitely-many-valued) function. The principal-value or principal branch of $\ln z$ is sometimes defined as $\ln r + i\theta$ where $0\le\theta\le2\pi$. However, any other interval of length $2\pi$ can be used, e.g. $-\pi < \theta \le \pi$, etc."

Bet that Murray Spiegel is feeling a bit silly now ... --prime mover (talk) 21:43, 15 December 2012 (UTC)

The drawback of such definitions is that $\log$ is either discontinuous (making theorems very awkward to state and susceptible to requiring the right choices among certain equally valid options, cf. a popular argument for proving $-1 = 1$ via square roots) or contains a branch cut on the negative real axis, refusing to define it (isn't that the principal branch, with $-\pi<\theta<\pi$ (both strict inequalities)?). In any case, Timwi's definition does not resolve the multifunction problem but rather relegates it to that of $\arctan$. In fact the canonical domain of $\log$ is a proper Riemann surface which can be imagined as some kind of corkscrewed complex plane (I'm sure a web search yields some nice pictures). Please do not sacrifice rigour for simplicity; $\log$ in a complex setting always needs at least some explanatory remark (and personally I detest $\ln$, but that's a different matter altogether). These intricacies probably lie at the heart of why complex analysis hasn't truly come off the ground yet on PW. --Lord_Farin (talk) 21:56, 15 December 2012 (UTC)

This is getting a bit polemic. I apologise if I stepped on anyone’s toes. My response was not intended to imply that anyone (or any books) were wrong. I was only talking about the theorem on this page. The theorem states that it assumes $x\in\R$ and the definition of arctangent on Definition:Arctangent, which is a single-valued function on the real numbers. (Other sources may define a different function and call it $\arctan$, but I’m going with this one.) The theorem states that it assumes this definition of $\arctan$ and that $x\in\R$. (The proof further states that it assumes $ y \in \R $, $ -\frac \pi 2 \le y \le \frac \pi 2 $, even though this is redundant because it follows from the aforementioned definition of $\arctan$ and the assumption on $x$.) With $y$ restricted in this way, I believe the proof to be correct for a version of $\ln$ defined as I did above. I did not realise that this is not how $\ln$ is defined on Definition:Logarithm. Next time I will take a closer look at the definitions. In the meantime, please decide whether you would prefer that we ① reword the theorem so that it specifically refers to this single-valued logarithm, ② create a new definition for this single-valued logarithm and call it something different than $\ln$, or ③ amend the theorem, the proof and Definition:Arctangent to work with multi-valued functions. — Timwi (talk) 22:27, 16 December 2012 (UTC)

(3) except you don't "amend the theorem", you expand it into the definition in the complex plane. Defining the arctan as a log of imaginary quantities is of limited value unless you have established the full complex-analytic definitions of these theorems in the first place.

Right, that's my lot for the night, it's getting late and I have an early start tomorrow. --prime mover (talk) 23:10, 16 December 2012 (UTC)