Talk:Exponentiation

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On 0^0 being correct[edit]

In the article we have

"According to Benson (1999), 'The choice whether to define is based on convenience, not on correctness.'"

Just what constitutes being "correct"? Correct based on what? Exponentiation starts with with x real and n a positive integer. I think we all consider this "correct". For positive x we use various means to generalize to negative exponents, rational exponents, and then real exponents. We also use exp(ix) = cos x + i sin x to generalize for x and n complex numbers with the proviso x != 0. Now, just what does it mean to have 3.7 factors, or sqrt(2) factors, or i factors for example? What makes these generalizations "correct"? We generalize the way we do because it is useful, convenient, satisfies the exponent laws, and "makes sense". In this light, how is incorrect? It is useful, convenient, follows the exponent laws, and make sense according to so many things. So I ask again, "correct" based on what?

We can use the exponent laws to narrow the possible choices to 0 or 1. The value 0 is useless. And as the article says, the value 1 is immensely useful. That makes it correct as much as any of the above generalizations.

Shouldn't we add something like the above? Perhaps a more concise version?

Betaneptune (talk) 18:18, 6 October 2016 (UTC)

The word correctness is used only in a quotation, and not elsewhere in the article. Nevertheless, in mathematics, a definition cannot be incorrect, unless it is self contradictory or induces a contradiction in mathematics. Generally, definitions are chosen for making statements as simple as possible. This is for this reason that 1 is not a prime number (otherwise, the uniqueness of factorization in the fundamental theorem of arithmetic would be difficult to express). Here, this is the same: is defined for satisfying the exponent laws and being continuous; there is only one function of x and y with y > 0, that satisfy both conditions (if continuity is dropped, there are many such functions). The problem, for is that continuity is impossible, as and This is for this reason that generally one defines when only integer exponents are considered, and one considers as undefined or undetermined, when exponents may varies continuously. Typically, in computer programming, if the type of the exponent is "integer", sand if the type of the exponent is "real" or "float" the result is NaN (Not a Number): Nevertheless, any definition for is correct, even But some definitions are more useful in some context, and the most useful definition depends on the context. I believed that this was clearly explained in the article. Maybe you have not read it with enough care. D.Lazard (talk) 21:02, 6 October 2016 (UTC)
(1) Then the article wasn't clear. I didn't see anything about making 0^0 anything other than 1 or undefined. I see no advantage to it being undefined. And why does x^y have to be continuous everywhere? What horrors ensue if it's discontinuous at (0,0)?
(2) Now consider 1^oo. Clearly this is 1. But when you consider 1^oo as a type of indeterminate form you can have lim(h->0) (1+h)^(1/h)=e. Would you then say that 1^oo equals e? Of course not. So why should limits mean anything for 0^0? (This is separate from the continuity argument.) You yourself said that definitions are chosen to make things simpler, as in 1 not being a prime number. Why not do the same for 0^0? Having 0^0 = 1 simplifies a lot of things.
(3) Every math book I've ever seen that gives e^x in summation format says it is good for _all_ x. This can only be true if 0^0 = 1. So you have a choice: 0^0 = 1, or you can undefine 0^0, which means you have to qualify the summation formula to be valid only for x != 0, and add a line saying e^x = 1 for x = 0. You can't have it both ways. Do you really want e^x have a two-line definition? And the same for other series starting with a constant term?
(4) As for the article, I think one could comment that the quote is ambiguous, because it is not clear what "correct" means. Perhaps the quote was taken out of context.
Betaneptune (talk) 04:05, 7 October 2016 (UTC)
The definition 00=1 is assumed in all cases where 00 occurs. The polynomial is supposed to be equal to , even for x=0. The discontinuity of yx around x=y=0 cannot be helped by undefining 00. However the undefiners insist. Bo Jacoby (talk) 03:05, 7 October 2016 (UTC).
I totally agree. Thank you. Betaneptune (talk) 04:05, 7 October 2016 (UTC)
This talk page is not for litigating what the conventions should be. In the article, we simply report what they are, in their existing multiplicity. Your remark above, and Bo's, are off-topic. --Trovatore (talk) 04:35, 7 October 2016 (UTC)
As Trovatore says Wikipedia is an encyclopaedia which describes what is out there. It is not a place for us to put in our own point of view and remove any other as in WP:NPOV. And by the way is another indeterminate form. For many purposes it would be clearly 1 but in other cases where we have and x and y are reals that happen to be one and infinity it would have no a priori value. What D.Lazard says above is a good description of the general situation. And by the way defining indeterminate forms to have definite constant values does give problems in calculus. I know calculus isn't being taught so much these days but it is still a very important part of mathematics. Dmcq (talk) 08:11, 7 October 2016 (UTC)
You just said that this is not a place to put our own point of view. Then you go on to add your own! I added my "POV" to explain why "correctness" in that quote is ambiguous, which is my primary point. One could simply add to the article: "But his criteria for what constitutes "correctness" is not given." That's not a point of view; it's a fact. If not, tell me what constitutes "correctness" in his quote and how it's obvious. Correct according to what?
I added my "POV" to support my proposal to add that one sentence. And [in response to your comment] you missed my point about form vs. value. As values, 0^0, 0/0, 1^oo are 1, "undefined", and 1, respectively. As indeterminate forms they are symbols that represent different types of limits. Not the same! Here's another example: 0/0. I think we'd both agree that this is undefined. Why? Because there is no unique solution to 0*x = 0. But we have lim(x->0) sin x / x = 1. This doesn't mean that 0/0 = 1. (!) So the limit and the value are different. With 1^oo, again the limit of an expression in that form and the value may be different. And so it is with 0^0. 0^0 = 1, but a limit in that form will vary. And the value of a function at a given point need not equal the limit at that point unless the function is continuous. But that's a property of the function, not the fixed number, whose symbol is used to represent the type of limit.
As Thomas says in his calculus book on p.654 (4th edition):
"Remark 2. Many people feel that the expression 1^oo cannot represent an indeterminate form. They say that one to any finite power is one, hence one to the power infinity is also one. But note, in the example above, that the base 1 + h is a variable which is not exactly equal to 1 except in the limit as h -> 0. There is a difference, in other words, between 1 ^ (1/h) and (1+h) ^ (1/h), even though both expressions formally become 1^oo as h -> 0."
So there is a difference between 1^oo the form, and 1^oo the value. And so it is with 0^0. And this is why 0^0 = 1 causes no problems in calculus.
And you say calculus isn't being taught much anymore? What? I beg to differ with that.
And D.Lazard saying 0^0 = Pi is correct? (1) It doesn't satisfy the exponent laws, which I think is a quite reasonable requirement. (2) There is no reason for it to be anything other than 1.
On D.Lazard's point about the implementation of 0^0 in computers, that is a decision made by the programmer -- not some ultimate truth or arbiter of "correctness".
Sorry if I went too far giving arguments for 0^0 = 1 in my original comment. I was just trying to demonstrate that 0^0 = 1 is a generalization or extension of the starting definition of x^n just like generalizing the exponent in x^n from integers, to rational numbers, to real numbers, and to complex numbers -- and just as valid.
Seems to me that after all this we can summarize it thusly: (1) I think we all agree (except for the remark of 0^0 = Pi) that the exponent laws must be satisfied. (2) There is disagreement on whether we need continuity at 0^0. (3) There is confusion between fixed values and limiting forms. OK, I'm guessing that Benson insists on continuity, but I don't see why that is unassailably "correct".
Anyway, aside from all the 0^0 = 1 bit, I don't propose to definitively state that 0^0 = 1 (even though it is!); I propose to add "But his criteria for what constitutes "correctness" is not given" as a comment on the quote by Benson. And I have presented my case.
Betaneptune (talk) 16:13, 7 October 2016 (UTC)
Saying 'But his criteria for what constitutes "correctness" is not given' would be us commenting on what a person said rather than an outside source doing it. He said what he said and thought it was good enough, I think we should just accept that. Dmcq (talk) 17:05, 7 October 2016 (UTC)
Betaneptune: The 0^0 section is a compromise. There is tension between what textbooks say about 0^0, and what they use+imply about 0^0. As you point out in your comment (3), textbooks give numerous formulas that assume+imply that 0^0 is 1. In other words, consistency requires that 0^0 is defined as 1. But many books don't explicitly say that 0^0 is 1. The 0^0 section is a compromise between two goals (a) consistency, and (b) reporting only what textbooks say (and not what they imply). But the two goals (a)+(b) are not compatible, which means that there's no easy way to fix the 0^0 section that wouldn't violate at least one of (a) or (b). MvH (talk) 02:46, 19 November 2016 (UTC)
I think the section is fixed. It reports the situation fairly well. Reading it again I thought it used to also mention that 0^0 was sometimes taken to be 0 in the complex domain, I wonder what happened to that. Dmcq (talk) 12:08, 19 November 2016 (UTC)
I agree that it reports the situation well. But given the tension between (a) and (b), we can't do better than a compromise. The line that Betaneptune quoted "The choice whether to define is based on convenience, not on correctness" is a compromise. We can't simply delete it because of goal (b). But at the same time, no mathematician should ever write a line like that, because even though it is technically correct, it is also misleading (every definition in math is based on convenience; singling out 0^0 is misleading). MvH (talk) 16:33, 19 November 2016 (UTC)
0^0 is singled out because there are a lot of situations where it makes sense to define it as 1 but it is not immediately apparent that it should be so and there is no logical reason arising from some some simple axioms to do so and there's good reasons to not define it like for 0/0. Dmcq (talk) 17:13, 19 November 2016 (UTC)
Regarding arguments for/against "0^0 = 1 in all cases", the point is that there are valid arguments (e.g. "a lot of textbooks don't explicitly define 0^0"), and there are invalid arguments, such as "the definition 0^0 = 1 is based on convenience rather than correctness" (that argument is without merit because the same is true for every definition in math).
But it may not be possible to delete every invalid argument and still have a balanced presentation of commonly held views. The current page is a good compromise.
Regarding axioms, historically, the definition of x^n is not based on any axioms, instead, the definition is motivated by convenience. Initially this meant: compactness of notation. Historically, x^3y^4 is simply an abbreviation of xxxyyyy.
Initially, exponents were at least 3 because x^2 is not shorter than xx. But of course, extending the definition to more exponents was a natural thing to do when that became useful, and extensions that satisfy more axioms tend to be more useful. But still, the rules/axioms come after the definition, not before. The definition is, as every definition in mathematics, based on convenience. Axioms only play an indirect role, in the sense that definitions that satisfy more axioms are more convenient. MvH (talk) 21:40, 19 November 2016 (UTC)

The Undefiner's POV assume the Trovatore conjecture: . Unfortumately the Trovatore conjecture is not main stream mathematics. Bo Jacoby (talk) 20:16, 7 October 2016 (UTC).

Bo Jacoby's conjecture is that Trovatore has conjectured something he has never written. Please stop such ridiculous assertions and stop to pretend that your personal point of view is main stream of mathematics. D.Lazard (talk) 21:35, 7 October 2016 (UTC)
I doubt that D.Lazard has read what Trovatore has written. Trovatore's POV is that is defined when the exponent is integer zero, but not when the exponent is real zero. That is . Bo Jacoby (talk) 06:34, 8 October 2016 (UTC).
Bo, I'm not interested in responding to how you characterize my remarks. But please don't do it here. It's off-topic on this page. --Trovatore (talk) 06:45, 8 October 2016 (UTC)
Trovatore, it is the topic of this talk page that once again a reader is confused that is sometimes undefined. Why shouldn't I comment here? Bo Jacoby (talk) 07:23, 8 October 2016 (UTC).
If you have suggestions as to how to help readers understand why many sources leave 00 undefined, and if these suggestions are themselves sourceable, then yes, that is on-topic here. --Trovatore (talk) 07:37, 8 October 2016 (UTC)
Could you do it without breaking the 2nd and 4th pillars in WP:5P please. Dmcq (talk) 09:15, 8 October 2016 (UTC)
When the topic is to define the exponential, then the sources that do not define the exponential are off-topic. Bo Jacoby (talk) 06:39, 9 October 2016 (UTC).
Please follow standard indenting and put your reply after what you're replying to thanks, see Help:Using talk pages#Indentation. Dmcq (talk) 11:29, 9 October 2016 (UTC)
Wikipedia:Silence_and_consensus. Bo Jacoby (talk) 05:54, 11 October 2016 (UTC).
No. --Trovatore (talk) 07:28, 11 October 2016 (UTC)
Wikipedia:Silence means nothing. Dmcq (talk) 09:37, 11 October 2016 (UTC)
The article needs to be cleaned from the off-topic undefining of 00. I leave it to you undefiners to contemplate that you got yet another user badly confused. Bo Jacoby (talk) 12:20, 14 October 2016 (UTC).
Perhaps Conservapedia would be more to your taste if you want an expurgated encyclopaedia, they like that sort of definiteness about the world. They might like that being put in there. Dmcq (talk) 15:04, 14 October 2016 (UTC)
The Trovatore conjecture fits nicely into the Conservapedia way of thinking. Bo Jacoby (talk) 08:42, 15 October 2016 (UTC).
I haven't been active on this article for a while, but it is clear that 0^0 is often considered undefined, so it is a violation of the pillars to assert that 0^0 is defined. The multiple-domain approach is WP:OR, but is accurate. As I see it, it's not saying that 0 ≠ 0.0, but that there are different definitions of the exponential function on different domains, and they may not have the same value at pairs which are in different domains. — Arthur Rubin (talk) 04:12, 10 January 2017 (UTC)

math to nowrap[edit]

Sometime in the the past year, all the {{math}} templates were changed to {{nowrap}}, for no apparent reason. I just changed all the {{nowrap}} templates back to {{math}}. — Arthur Rubin (talk) 04:24, 10 January 2017 (UTC)

The status as of the beginning of 2016 was that some where {{math}}, and none were {{nowrap}}. An anon added a lot off {{math}}s on Feburary 2, 2016, which was promptly reverted. However, no consensus or attempt at consensus is evident in the edit comments or on this talk page. I think the status quo ante is better represented by use of {{math}} than by {{nowrap}}. — Arthur Rubin (talk) 04:39, 10 January 2017 (UTC)

Missing comments?[edit]

There appear to be no comments between 2011 and 2015. This seems unlikely to me; perhaps they were archived to a non-indexed subpage.... — Arthur Rubin (talk) 04:43, 10 January 2017 (UTC)

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Merge with power function?[edit]

Partial merge from Power function to Exponentiation. power~enwiki (π, ν) 19:51, 3 November 2017 (UTC)
The following discussion is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

The power function article seems to cover the same topic as exponentiation, but includes much less information. Is there a reason to have two separate articles? 2601:647:4D03:3CA7:D4F2:9F2A:1837:9270 (talk) 12:42, 1 September 2017 (UTC)

Well, no, it's not really about exponentiation. Exponentiation is a two-argument function; the power-function article is about one-argument functions obtained holding the exponent fixed.
That's a plausible topic for an article, though it could also plausibly be merged into power law. A separate question is whether "power function" is really a standard name for it; I'm not sure that it is.
Anyway, I'm not sure what is the best thing to do with the article (merge into power law, move to some other name, or conceivably even delete — I don't think deletion is a very likely option but someone could argue for it). But merging it here doesn't seem right to me. --Trovatore (talk) 20:37, 1 September 2017 (UTC)
  1. Delete Power function (PF), after merging any marginal content there, which is not yet already contained in here. I do agree on exponentiation being a binary function in some way, but the current content here covers already an overwhelming part of the whole PF-article. I also share the doubts about PF being a genuinely accepted term, and I would vote for splitting this here article in three or even more (getting rid of 0^0!), if it had not its current valuation. Purgy (talk) 10:00, 2 October 2017 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

Structure on complex powers of pos. reals[edit]

@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 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 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 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 instead of focusing on 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 ; 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 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 to
-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 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 : 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 a matter dealt with already in real context, so the fist equation in the first math-line of the article, stating for complex is fine. However, the second equation I consider to be a theorem, requiring a proof that depends on the definition of 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(!) via wlog, and that
- the proof of EF then depends on the definition of and of the trigs. I also mentioned that
- the claim 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)