Talk:Exponentiation/Archive 2018

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Structure on complex powers of pos. reals

@Danstronger: I admit that the examples employ Euler's formula, so I postponed them in a new section, just to start with the definitions (power series, limit), where the formula drops out effortlessly, and needs no anticipation.

Furthermore, I think
- this is an article about exponentiation, so the trigs do not deserve a header at this level,
- this is also not an article about Euler's formula, having a main-link, so I'd rather reduce than expand its prominence,
- I left the header for imaginary exponents for the main-link and removed the periodicity header now, since you removed it too,
- I resolved the doubly crop up of Euler's formula and the trigs by referring to them (I believe in the series definition to be slightly more important).
- your final caveat for complex bases belongs to the top, imho.

I gave my intentions a shot, feel free to improve on them. :) Purgy (talk) 11:08, 29 November 2018 (UTC)

I was thinking the thing about $(b^{z})^{w}$ fits best under "properties" (where it would be written with e instead of b), but I don't feel strongly about it. I also think the formula $e^{x+iy}=e^{x}(\cos y+i\sin y)$ should go as high in the section as possible, preferably in the intro. It might be good to have all the definitions up front (b^z, e^ix, e^(x + iy)), and then the rest of the section can be examples, derivations/illustrations, and properties. The main-link Euler's formula should go with the section called "purely imaginary exponents"; Euler's formula is the statement of what e^z is when z is pure imaginary. Maybe the stuff in "Trigonometric functions" should just be part of "properties" (or remove it completely)? If "trigonometric functions" is a section, I don't think it needs a main-link. Danstronger (talk) 04:01, 30 November 2018 (UTC)

Euler's formula as normally stated is with an imaginary exponent not complex numbers in general. I do think though the trigonometry section should be removeds, Euler's formula is described in the previous section well enough. I'll go and do that' Dmcq (talk) 11:50, 30 November 2018 (UTC)

@Danstronger, thank you for discussing your views! Below I'll try to present my POV.

- I do not consider this a property, because the general case of base $(b^{z})$ is already excluded by the header. So this is, imho, more of a caveat than a genuine property, and I plea for leaving it upfront.

- I truly admire Euler, and not only for this formula, and I am even convinced him being prominently important within the complex numbers, not only for this deep identity, but already for being one of the first to perceive the troubles caused by stating $i={\sqrt {-1}},$ instead of focusing on $i^{2}=-1,$ but I like to see the stand-alone version of complex numbers at the forefront of defining exponentiation, considering both the Cartesian as the polar representation as problematic, and to be used with utmost care. As a primary aside: the first needs the Cauchy-Riemann equations kept in mind (F(z) vs. f(Re, Im)), the second has universal coverings in their backdrop. So I perceive no urgency with an early appearance of Euler's formula, necessarily in need of both two representations.

- Well, the trig-section got rid, we will see whether recovering the connection of adding angles and exponents will survive, and then I got rid of the header of imaginary exponents as well.

I think your concerns are either removed, or addressed above. :) Purgy (talk) 13:13, 30 November 2018 (UTC)

Thanks. I think the explanation that z can be written as x + iy was originally intended as a quick intro to complex numbers, since they hadn't come up in the article yet, rather than as an endorsement of cartesian coordinates. But perhaps it's best to just assume readers have basic familiarity with complex numbers; if not they can click the wiki-link. I left the (b^z)^w caveat at the top, but I tried to make it more concise. I also noticed that there was stuff under "Taylor Series Definition" that didn't really fit under that heading, so I removed that heading and the ones for "Limit Definition" and "Purely Imaginary Exponents". I was thinking maybe there were too many headings anyway. But perhaps now the section appears to ramble? I also fleshed out the argument about the trig sum formulas and made each example one line; they seem clearer to me that way. Danstronger (talk) 03:34, 1 December 2018 (UTC)

Sorry, but I have some objections to your last edits.
- Removing the hint to the definitions allowing for the second equality leaves the claim unfounded. I think the hint is necessary.
- The same holds for explicating the restriction to define exponentiation just for e (no loss of generality ...).
- My reason for breaking down the examples to shorter lines was to allow for a better layout wrt varying screen widths (e.g. scroll bars, whitespace with pic, ...). This also holds for your fleshing out the trig sums. (I wouldn't have inserted the cos-sum, additionally.)
- I had the "im. exp."-header removed myself, but I miss the structure showing the two "characterizations" of exponentiation, I vote for keeping the headers "exp. def." and "lim. def.", they ease the reading, one may skip the second, ... (Removing the cursory introduction of complex numbers is fine with me, I linked "compl. n." exactly for this reason at the beginning.)
- Removing the hint to Euler's formula needing different proofs, depending on the characterizations, makes the article more vulnerable, so I vote against.
I'll wait for your ideas before being myself bold again. Purgy (talk) 09:01, 1 December 2018 (UTC)

I think I'll move the properties section above the examples section. I think it was a mistake to remove the header for purely imaginary or whatever it was as properties is just the wrong header for including a real part. Dmcq (talk) 18:27, 1 December 2018 (UTC)

I added headers for Euler's Formula (including the stuff about $e^{x+iy}$; hopefully it fits better there than under properties), Limit Definition, Periodicity (the only property left), and Examples, in that order. I think examples at the end makes sense. Danstronger (talk) 22:35, 1 December 2018 (UTC)

I think the second equality is the definition of b^z; I changed it to say "b^z is defined as". I don't think "without loss of generality" is really the right concept here. b^z is defined in terms of e^z; e^z is a special case, defined through the exponential function. I don't think the current material on the limit definition really constitutes another proof of Euler's formula. The argument that 1 + i pi /n "approaches" the appropriate point on the unit circle is hand-waving. Since we're not really offering a choice of proofs, I thought it was ok to just give the proof based on Taylor series and link to characterizations of the exponential function. About the trig, I thought if we mention the sum formulas we might as well give that two-line argument for them. Anyway Dmcq has that stuff out altogether, which is also fine with me. I think the examples look better as one line each on a wide screen, and equally bad on a short screen, but feel free to change it back if you think it's better the other way; I won't change it again. I'm definitely open to headings being added back in, but I don't see a clear way to do it now. Danstronger (talk) 20:36, 1 December 2018 (UTC)

Yes I think that looks good. Thanks. Dmcq (talk) 00:02, 2 December 2018 (UTC)

There is not much I object to, but definitely

- the second equality cannot serve as a definition, it is in fact a theorem, when taking the series as definition for $e^{z}.$ I tried to mention this fact with stating that this series would allow for this equality, and only therefore it may be used "without loss of generality" to expand $e^{z}$ to $b^{z}.$

-As other, lesser points, I'd rather leave the header as "Series def." of exponentiation and just link "EF", when it drops out of the series def. It is an article about "exponentiation", not about single, important connections. I do like having both contrasting definitions, even when one is just hand waving; this is a fruitful view on the topic, imho. In the light of these two definitions for $e^{z},$ and the importance of EF, the remark about the EF requiring different proofs, depending on the setting of premises for both, trigs and power, might be even more necessary for a consistent article. BTW, Purgy (talk) 09:26, 2 December 2018 (UTC)

I'm not sure I follow you. As far as I can see the limit definition is the definition using the limit form definition of e^x. I'm not sure how it can be considered a theorem unless you are starting from another definition like the series one.

If the bit that :$e^{z}=e^{x+iy}=e^{x}\cdot e^{iy}$ is moved to the introduction section that will provide a rationale for dealing with the e^iy part separately and then Euler's formula can deal with that. The exponential function is dealt with in its own article but I think it is okay to then use two definitions to show how Euler's formula works. Dmcq (talk) 10:18, 2 December 2018 (UTC)

Sorry, I was yet unable to be sufficiently explicit about my concerns. Another shot: I consider the equation $b=e^{\ln b}$ a matter dealt with already in real context, so the fist equation in the first math-line of the article, stating $b^{z}=(e^{\ln b})^{z}$ for complex $z$ is fine. However, the second equation $(e^{\ln b})^{z}=e^{z\ln b}$ I consider to be a theorem, requiring a proof that depends on the definition of $e^{z}.$ In my (edited out) formulations I mentioned that

- the given definitions (series and limit) allow for this theorem, and that

- this theorem allows to define(!) $b^{z}$ via $e^{z},$ wlog, and that

- the proof of EF then depends on the definition of $e^{z},$ and of the trigs. I also mentioned that

- the claim $e^{x+iy}=e^{x}\cdot e^{iy}$ is warranted within the series definition by the commutativity of the complex multiplication and the absolute convergence of the series.

I think these comments could be -in decreasing urgency- meaningfully incorporated into the article, improving its consistency. Purgy (talk) 12:15, 2 December 2018 (UTC)

No that is a definition not a theorem because complex powers of b are not defined except by complex powers of e. It can also be considered a requirement for powers to a complex exponent to be natural extension to that for the reals. At most the worry is whether the definition doesn't give a well defined value or breaks some other reasonable expectation for exponentiation. Also it is not Wikipedia's job to prove things, that should be left in general to citations. Dmcq (talk) 13:33, 2 December 2018 (UTC)

OK, I understand that "theorem" is the wrong term for something making a definition "consistent". As it stands now, I perceive "some other reasonable expectation"s swept under the rug. Should one better talk about holomorphic continuation? I miss why omitting all these finesses yields a better article, but I won't bother anymore. Purgy (talk) 08:00, 3 December 2018 (UTC)

Problem with order of exposition

In the section Complex exponents with a positive real base, $b^{z}$ is defined using the definition of $e^{z}$, but $e^{z}$ isn't defined until later. I tried editing the page to first give the series definition first, which would then make sense. However my edits were reverted, on the grounds that the series definition isn't the only definition of $e^{z}$. And while that is technically true, it sort of misses the point, which is that *some* definition of $e^{z}$ should be given before it is expanded to the general definition of $b^{z}$. — Preceding unsigned comment added by Hatsoff (talk • contribs) 15:39, 21 December 2018 (UTC)

I understood your concern, and I have edited the page accordingly (before reading your post). However, I agree that the section deserves to be restructured into a subsection "Complex exponential function", including as subsubsections the various equivalent definitions, preceded or followed by a subsubsection "Properties", and then a subsection "Arbitrary positive real basis. This requires more work, and I'll not do it for now. D.Lazard (talk) 16:13, 21 December 2018 (UTC)

Okay, cool, I agree it should be cleaned up a bit more, but for now it's good enough. Thanks! Hatsoff (talk) 16:37, 21 December 2018 (UTC)

Plus I'd remove "unless z is real or an integer". If z is a complex number it is still a complex number even if the imaginary part is zero, this is how the modern definition of functions work. Dmcq (talk) 21:07, 21 December 2018 (UTC)