# Talk:Euler's formula

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## Citation needed for proofs

There has been questions asked about the proofs so a citation is needed on the ones that are not yet cited. The proofs should be similar to ones in books using the same general principles. Please no removing the citation needed tags with 'And that citation tag is not needed for a simple and explicit mathematical proof' and then going and providing no citation. Wikipedia is not a place for people's own proofs. Dmcq (talk) 23:09, 19 April 2011 (UTC)

You know, that is really silly. Being competent at mathematics, you should know that any proven theorem becomes a tautology. Are you saying that all tautologies need a reference to a previous use of it to be eligible for Wikipedia? While the power series proof is most common in textbooks, it is not alone and these textbooks not only have alternative ways of looking at it in the text proper, they put in problems at the back of the chapters for such a purpose. Does such a problem (unsolved in the textbook) qualify as a reference?
I have seen both the "calculus proof" as a problem in my old calc book (Seeley, Calculus of One and Several Variable) and a proof involving 2nd-order diff eq in EE texts (also Kreyszig, Advanced Engineering Mathematics).
The power series proof is less complete as a proof because it requires knowledge of the Maclauren series as a starting point. To be as complete, starting from the same axioms (say, the rules of differentiation in calculus and that i is a constant such that i 2=-1), the power series proof requires many more steps.
To imply that it's just a personal proof is either really dumb or at least ignorant.
You would make Wikipedia less useful to readers for the sake of your legalism. A shame. 70.109.189.158 (talk) 02:45, 20 April 2011 (UTC)
I think the point is that a proof just doesn't belong in the article if it's not in the literature and verifiable. Readers and editors shouldn't be in the business of verifying proofs themselves. As for your extrapolation to "all tautologies", I don't know what means here, so I take no position. And there's really no need for proofs reviewed here to be "complete"; verifiability in a reliable source is a good substitute for that. See WP:V to understand more about "verifiability, not truth". Dicklyon (talk) 02:55, 20 April 2011 (UTC)
If it's so simple that any of the competent editors here know that it is valid, what's the point of seeing it in a textbook? In fact, should every "proof" in a textbook be sufficient for Wikipedia? You and I both know, Dick, that there are plenty of proofs in EE texts regarding sampling that are not considered acceptable by mathematicians because they leave the Dirac delta function naked (without the surrounding integral) and do manipulations with it. What you are saying is that Wikipedia should not be helpful in answering this question "Why?" or "How?" for those whose concepts of beginning calculus is working (they know how to take derivatives) but of latter concepts (like Maclaurin series or of complex variables) is weak. You're telling them that they just have to accept the Maclaurin series proof, even though it is more complicated and unnecessary. 70.109.189.158 (talk) 03:05, 20 April 2011 (UTC)

A citation isn't just for readers who think "I don't believe this statement -- Is it really true?" It's also for readers who think "I want to read a more thorough discussion of this point." So yes, we should have them, even on some statements that are "obviously" true.

As I discussed above, the calculus proofs require the prerequisite claim that the complex exponential function obeys the same calculus identities as the real exponential function. It's easy to just assert, "the complex exponential function obeys the same calculus identities as the real exponential function". If we say that, readers will say, "OK that's plausible". Then we show them the "proof" and the readers will say, "OK that's plausible". I guess that's a good enough reason to put it in the article. The fact that textbooks use this "proof" is also good enough reason to include it in the article. But we should not pretend it's a real "proof", because we haven't actually shown that the prerequisite claim is true. And there's no easy way to do so. Except using Maclauren series! :-) --Steve (talk) 04:47, 20 April 2011 (UTC)

Yes, there are good proofs and bad proofs in sources. We should endeavor to choose to base the article on good ones, and cite those, not to make up our own and claim they're good, or that they're better than what's in sources, or that they're so obvious that no verification is needed. We go through this issue all the time, and the answer doesn't change: if someone requires a citation for verifiability, they should get one within a reasonable time, or they're justified in removing the material. The burden falls on whoever wants to be helpful, or whoever wants the material to remain. I spend my most productive editing time find and citing sources, because that's what turns ephemeral junk into a lasting piece of wikipedia. Dicklyon (talk) 06:25, 20 April 2011 (UTC)
If the statements about Seeley, Calculus of One and Several Variable) and a proof involving 2nd-order diff eq in EE texts also Kreyszig, Advanced Engineering Mathematics just have page numbers added and really are similar then they can be turned into references quite easily. That's all that's being asked for. If you know where things are described why not help an article by giving the information? Using the template of another citation in the text would format it but people in Wikipedia are happy enough to format things like that if the basic information is just put in as text. Dmcq (talk) 11:01, 20 April 2011 (UTC)
I really dislike the deletion of the calculus proofs since they were in my opinion much more elegant than the proofs that remain in the article (if you can call what remains of the proof from the limit definition a proof at all). I've looked around a bit for references for the calculus proofs, and found one similar proof from the textbook of Gilbert Strang which is available for free online although I prefer the proof that was in the article before. I am planning to put the Strang proof into the article as it is quite short.Holmansf (talk) 20:39, 7 February 2012 (UTC)
I was unable to find the proof in the citation you provided. In fact I found the first proof using a series on page 389 of Chapter 10 'Infinite series'. I shall mark that proof with not in citation given and remove it again after a few days unless a proper citation is given. Citations should be much more specific than what was given. Dmcq (talk) 23:12, 7 February 2012 (UTC)
You didn't look very hard. It is further down the page on p389.Holmansf (talk) 03:37, 8 February 2012 (UTC)
I did look hard. That's how I found that page when you had not supplied it out of the 38 megabytes. The second proof was not marked out in bold like the first one and why should I be expected to search for a second proof? At least one can be expected to search a single page. See some other citations too on supplying year of publication, publisher and isbn, they are not so important as the page numbers but putting any of those in that one can helps, e.g a second issue can change page numbers. Dmcq (talk) 04:34, 8 February 2012 (UTC)
I think the justification of $d/dx e^{ix} = i e^{ix}$ from the power series should remain in the calculus proof. In fact one editor above complains at length about the absence of this justification in the previous calculus proofs. If it is a reference you are concerned about, I believe the Arnol'd book covers this (p164 when A is identified with multiplication by i: R^2 -> R^2). Commments? Holmansf (talk) 16:28, 8 February 2012 (UTC)
Putting the power series in makes the proof clumsy: then it makes more sense and is more elegant to stick with the power series and leave calculus out of it. If you want a justification of that statement, then start from the definition of the exponential function as the solution of the differential equation df/dx=f with the initial condition f(0)=1. — Quondum 16:57, 8 February 2012 (UTC)
The source just started with the derivative of eix, I don't see why we need try making it any more specific. In fact this might be better for a proof where the exponential function was defined by the derivative and value at zero rather than a series. Dmcq (talk) 18:08, 8 February 2012 (UTC)
Then, to waylay the objection, perhaps say "Another proof[7] starts with deix/dx=ieix and is based on the fact that..."? — Quondum 18:31, 8 February 2012 (UTC)
My opinion is unchanged. How is it possibly better to say "Now from any of the definitions of the exponential function it can be shown that eix when differentiated gives ieix" or "Another proof starts with deix/dx=ieix " than to actually give a three line entirely self contained proof of this statement which starts from one of the two clear definitions given in the article. Especially when the lack of such a proof is *precisely* what began this whole discussion of the calculus proofs. Holmansf (talk) 18:48, 8 February 2012 (UTC)
Also, it does not work to take df/dx=f with the initial condition f(0)=1 as a definition of the complex exponential as I explained here. Holmansf (talk) 18:56, 8 February 2012 (UTC)
I'll accept that the definition of ez as the solution to a differential equation is not great (it depends on the requirement of being complex analytic/holomorphic as you have argued, and in a sense goes beyond simple calculus, meaning differentiation). My question would then be what this proof adds, since any route seems to depend on some more sophisticated result, with the exception of the use of the Taylor's series for ez. And using that makes it less elegant than the Taylor's series proof. It also relies on knowledge on the derivatives of the real functions cos and sin (even though it avoids their series'), so that is lots of premises. So it does not live up to its promise of elegance. — Quondum 19:29, 8 February 2012 (UTC)
You don't have to do the whole business at once. Only eix is needed to start with and then one can define ex+iy=exeiy and show that works after proving Euler's formula for the pure imaginary case. The main point though is the source starts with that and not with a series and the series case is better dealt with by the first proof just as in the book. Agree overall though is it doesn't really add that much and is it actually of any note? Dmcq (talk) 21:06, 8 February 2012 (UTC)
For me, the proof is not complete unless it connects one of the given definitions to the formula. I could understand objections if doing that in a rigorous way required a lot of extra steps, or knowledge well beyond the presumed level of the intended audience - but it doesn't in this case. Further, this same simple calculation differentiating the power series shows why eAx is defined the way it is for linear operators A, and how that relates to the solution of certain systems of ODEs. Thus it does in fact give extra insight. Also arguments that the series proof is better than the calculus one with this addition are simply based on taste, and I just don't agree. Holmansf (talk) 21:33, 8 February 2012 (UTC)

I'm coming to the conclusion the proof from the derivative should just be removed. There is an essay about proofs at Wikipedia:WikiProject Mathematics/Proofs which expands the bit in WP:MOSMATH#Proofs. I don't think we can really argue that the proof is notable in any way and it is redundant. Dmcq (talk) 23:40, 8 February 2012 (UTC)

It doesn't seem like we're going to come to an agreement about this. I've asked for additional comments over at the Mathematics Project talk page. Holmansf (talk) 00:00, 9 February 2012 (UTC)

As an outside opinion, having read the comments above: I think that three proofs is too many, and I think that the "limit" proof in the article currently is the worst-presented. It would be ideal to have a reference for each of the proofs that remains in the article. — Carl (CBM · talk) 00:46, 9 February 2012 (UTC)

I think all three proofs are sufficiently unique to be included. I'm just coming into this so I'm not completely familiar with how the article has been changed recently, but the current version appears to a good example of how WP should handle proofs. I think the series proof is the most standard and should be included for that reason as well as it's a good illustration of the use of power series. The proof using the limit definition of exp is actually an application of Euler's method to solve the differential equation, and it's good to have a proof that doesn't make assumptions about the derivative. It's really a sketch rather than a proof since it leaves out most of the detail, but that's appropriate in an encyclopedic treatment. The animation for it is an added bonus. The calculus proof isn't familiar to me, but it's concise, referenced, and has an elegance that is lacking in the other two.--RDBury (talk) 01:49, 9 February 2012 (UTC)

To cut through all the blether above, Wikipedia is not a place to go to for mathematical proofs. It's an encyclopedia and not a mathematics text book. If it's proofs you want, go to ProofWiki or something like that, whose raison d'etre is to provide proofs. Them's the rules, unfortunately, stupid as they are. --Matt Westwood 06:23, 9 February 2012 (UTC)
I have seen a few people say things like this, but actually as far as I can tell there is no agreement on this subject. The essay Wikipedia:WikiProject Mathematics/Proofs linked above seems to support including short proofs. Indeed, the "examples" given there that I looked at all included multiple proofs (sometimes sketches really) of the same result. Holmansf (talk) 13:22, 9 February 2012 (UTC)
The difference between ProofWiki and Wikipedia is that the former does not have a draconian deletion policy. If it's a result, it's in, however trivial. Some don't like this, they think trivial is bad. My advice to them is to stay with Wikipedia, to which they are clearly more well suited. --Matt Westwood 18:24, 9 February 2012 (UTC)
I'm not sure "trivial" is really the criteria for deletion. For instance one of the articles which is supposed to provide an example of how proofs work on wikipedia provides two separate proofs of the Difference of two squares formula $a^2 -b^2 = (a-b)(a+b)$. This is probably the most trivial result for which I've ever seen someone actually bother to write a proof ... but yet that is an "example" of how proofs should work on wikipedia. In fact it seems to me based on this experience that there is no concrete policy at wikipedia on mathematical proofs, and what's in or out is basically up to the preference of individual editors. Holmansf (talk) 18:42, 9 February 2012 (UTC)
The article already contains two proofs and you are trying to stick stuff into the third proof which isn't in the original source so as to make it complete as you see it. What that essay says is 'The role of proofs, which may be short but correct arguments, or sketches of longer arguments serving more as a map of complete proofs, is to support the "survey" and "reference" ambitions.' Dmcq (talk) 13:53, 9 February 2012 (UTC)
It seems to me that more editors who have responded support inclusion of the calculus proof than deletion. I'm willing to leave it as is to avoid continuing this argument. Well actually, I want to slightly change some of the wording. Hope that's okay. Holmansf (talk) 14:59, 9 February 2012 (UTC)
Can't say I like the way you've turned it into a deus ex machina with no explanation of why the statement is a valid possibility. Dmcq (talk) 15:30, 9 February 2012 (UTC)
Is it the initial statement $e^{ix} = r(\cos(\theta) + i \sin (\theta))$ that you're concerned about? Holmansf (talk) 16:01, 9 February 2012 (UTC)
Yes. The original source and what was there both gave reasons for that being okay. There was a complaint on this talk page which was then reverted by someone misunderstanding it when it was presented without an explanation ' recently added] "proof" by calculus is truly ghastly. It seems to assume its conclusion as a premise'. So I think something like that is needed.Dmcq (talk) 16:36, 9 February 2012 (UTC)
Is that better? I don't like the wording in the source because it is technically incorrect (zero is not e^{ix} for any x), and I was just trying to shorten the previous wording in the article which seemed too long to me. Holmansf (talk) 17:04, 9 February 2012 (UTC)
From my point of view the 'deus ex machina' is saying the derivative of e^{ix} is i e^{ix}. ;) Holmansf (talk) 17:06, 9 February 2012 (UTC)
No what you changed to still had the problem so I've reverted. 'is a complex number and all complex numbers can be expressed as' does not imply can be zero and is how I've summarized what I believe the author meant. The book doesn't work out the derivative of eix.so I don't see why we should. In fact in Characterizations of the exponential function they have that as a starting point but don't prove its equivalence to the others which is a bit bad so you might like filling in something there if it worries you. Dmcq (talk) 17:54, 9 February 2012 (UTC)
What specifically was wrong with my previous wording? Holmansf (talk) 18:16, 9 February 2012 (UTC)
It is true but still I think rather liable to misinterpretation as assuming the result, and a person did exactly that with something very similar. Seeing that misunderstanding such a short time after it was put in by someone who is no slouch in the maths department is what made me be very careful about trying to phrase it properly. Saying that it is a complex number and all complex numbers can be expressed that way gets over that hump. Dmcq (talk) 18:33, 9 February 2012 (UTC)
That is not what it said when this person entered the "ghastly" comment. What I am asking about is this:
Another proof[7] begins by expressing the complex number eix in polar coordinates as
$e^{ix} = r (\cos(\theta) + i \sin(\theta))\,$
for some r and θ depending on x. Holmansf (talk) 19:03, 9 February 2012 (UTC)
I don't see any response on why this wording is inappropriate. I am planning to revert to this wording unless I see a response. Holmansf (talk) 03:23, 11 February 2012 (UTC)
This suggested wording is not quite reverting it to what Dmcq changed it from. However, your new suggested wording "... expressing ... in polar coordinates ..." goes a long way towards preempting the potential confusion (due to similarity with the form of the conclusion) that I, for example, experienced (admittedly being tired) when I read "Now we must have ...", and is neater. — Quondum 05:48, 11 February 2012 (UTC)
I've put in 'polar coordinates' instead as suggested, that does look better than having the formula. Dmcq (talk) 10:32, 11 February 2012 (UTC)
It was not new. That was a direct quote from what Dmcq reverted. I'm reverting. Holmansf (talk) 14:30, 11 February 2012 (UTC)
The big problem is not whether polar coordinates is said in words or the formula given, it is making it explicit that the result is not being assumed and that's what I think you remove. Dmcq (talk) 17:02, 11 February 2012 (UTC)
Hmm, I really don't see how this is better than what I had which was more concise. I'll let you prevail however in the interest of compromise. Holmansf (talk) 23:50, 11 February 2012 (UTC)
The way I see it a person could say "but isn't that what you're trying to prove" if you don't point out why it isn't. And the reason it isn't is because any complex number can be put in polar coordinates. Dmcq (talk) 00:02, 12 February 2012 (UTC)
Also, I don't see the derivative of eix discussed anywhere in Characterizations of the exponential function. Holmansf (talk) 18:31, 9 February 2012 (UTC)
The real function is defined that way in characterization 4 and the next section talks about using the characterizations in wider domains. Dmcq (talk) 18:36, 9 February 2012 (UTC)

────────────────────────────────────────────────────────────────────────────────────────────────────I think the discussion here is getting too hung up on what the assumptions should be. It's a general problem with doing proofs on an encyclopedia that, since each article is independent, you don't get to control which facts are currently "known". Actually, if you want to be strict logically then really there is no need for a proof, simply define the value of exp(x+iy) to be exp(x)(cos(y)+isin(y)); you then just have to verify that the values match on the reals, the usual laws of exponentiation, and the new function satisfies the Cauchy-Riemann equations for differentiability. But this wouldn't give readers any insight as to why the function is defined this way.

I think for this article we should assume that readers of this article have basic calculus, know a minimal amount about complex numbers, and may be interested to know why this crazy idea of connecting trigonometry and exponentiation suddenly makes sense when when you throw i in. I don't think they are interested in being 100% mathematically rigorous or worried about whether a particular function is continuous and differentiable. All three proofs have issues if you want to be completely rigorous, but being overly particular on rigor will lead to something like Bourbaki, mathematically correct but only a few grad students will bother to read it.--RDBury (talk) 23:06, 9 February 2012 (UTC)

I appreciate your point, but this is a long way from Bourbaki. Holmansf (talk) 03:19, 11 February 2012 (UTC)

## Critics on Proof Using differential equations

The Proof using differential equations start with the derivative of $\frac{d}{dy} e^{ix} = i e^{ix}$. And then it is shown that $\frac{d}{dy} f(x) = i f(x)$. But how then the conclusion is $f(x) = e^{ix}$ ? Wisnuops (talk) 06:40, 16 January 2012 (UTC)

As far as I can tell, the paragraph at the end of the section Using differential equations explains this. If it is not adequately clear from the explanation (from that they are both the solution to the same differential equation for all values of x and are equal at at least one value of the domain) that they must be equal, is there some way in you could suggest improving the explanation so that it would be easier to understand? — Quondum 08:46, 16 January 2012 (UTC)
I removed the proof as that little point requires some work one would only do long after Euler;s formula anyway! AN it wasn't cited. The calculus proof is better if you want this type of approach. Dmcq (talk) 04:56, 8 February 2012 (UTC)

## Euler's own proof

Would there be any interest in including Euler's own proof? It is shorter, and more convincing, than the three proofs currently appearing in the article. Furthermore, there are modern references so the material is easily verifiable. Tkuvho (talk) 14:23, 9 February 2012 (UTC)

Enough with the dangling of carrots. With all those claims, let's see it. What better material could there be for an encyclopedia article on the topic? — Quondum 15:23, 9 February 2012 (UTC)
Sure, of course. Although the article currently claims that Euler used the infinite series proof: "his proof was based on the infinite series of both sides being equal." Is this statement wrong? Holmansf (talk) 15:28, 9 February 2012 (UTC)

## e to the pi i

Somewhere in the introduction the familiar (and amazing!) expression e**pi*i = -1 needs to be shown. Casey (talk) 22:51, 25 September 2013 (UTC)

## "j" vs "i"

The "3-D" graph uses "j" (as in electrical engineering) rather than "i" (as in mathematics). Even though most perusers of this article will recognize the difference/sameness, to avoid confusion either a graph using "i" or an explanation on the graph itself is desirable. Casey (talk) 22:49, 25 September 2013 (UTC)

## A proof based on De Moivres formula

Eulers formula, for real x, may be obtained from De Moivres formula, for integer n,

$(\cos_\theta + i \sin_\theta)^n = \cos_{n \theta} + i \sin_{n \theta}$

Let $\theta = \frac{x}{n}$, and take the limit as n tends to infinity;

$\lim_{n \to \infty}((\cos_{\frac{x}{n}} + i \sin_{\frac{x}{n}})^n = \cos_{\frac{n x}{n}} + i \sin_{\frac{n x}{n}})$

Using the power series expansions,

$\cos_{\frac{x}{n}} = 1 + \frac{0}{n} + \frac{?}{n^2} + ...$
$\sin_{\frac{x}{n}} = 0 + \frac{x}{n} + \frac{?}{n^2} + ...$

gives,

$\cos_{\frac{x}{n}} + i \sin_{\frac{x}{n}} = 1 + \frac{i x}{n} + \frac{?}{n^2} + ...$

In the limit of the binomial expansion of $(\cos_{\frac{x}{n}} + i \sin_{\frac{x}{n}})^n$, it can be shown that the sum of all the terms arising from $\frac{?}{n^2}$ and higher power terms will go to zero as n goes to infinity. So,

$\lim_{n \to \infty}((\cos_{\frac{x}{n}} + i \sin_{\frac{x}{n}})^n) = \lim_{n \to \infty}((1 + \frac{i x}{n})^n) = \cos_x + i \sin_x$

Two pathes to the result are possible from this point;

Complex numbers are a field. Direct expansion

Consider the function,

$f(y) = \lim_{n \to \infty}(1 + \frac{y}{n})^n$

where y is a real number. Applying the change of limit variable $m y = n$ gives,

$f(y) = \lim_{m \to \infty}(1 + \frac{1}{m})^{m y} = e^y$

$e^y$ may be represented as a convergent power series,

$e^y =\sum_{k=0}^\infty \frac{{y}^k}{k!}$

The complex numbers form a field, so a power series may be constructed from complex numbers. So $e^z$ may be defined for complex z. f may also be expressed as a power series, which can be generalized to complex numbers. Therefore the coefficients must be equal and the expressions equal for all complex z. So,

$e^z = f(z) = \lim_{n \to \infty}(1 + \frac{z}{n})^n$

Expanding the limit using the binomial expansion,

$(1+\frac{z}{n})^n = \sum_{k=0}^n \frac{n!}{k! (n-k)!} (\frac{z}{n})^k$

Which may be re-arranged as,

$(1+\frac{z}{n})^n = \sum_{k=0}^n \frac{z^k}{k!} \frac{n!}{n^k (n-k)!}$
$= \sum_{k=0}^n \frac{z^k}{k!} + \sum_{k=0}^n \frac{z^k}{k!} (\frac{n!}{n^k (n-k)!} - 1)$

The limit of the second term can be shown to tend to zero as n goes to infinity,

$\lim_{n \to \infty}(\sum_{k=0}^n \frac{{z}^k}{k!} (\frac{n!}{n^k (n-k)!} - 1)) = 0$

so,

$\lim_{n \to \infty}(1+\frac{z}{n})^n = \sum_{k=0}^\infty \frac{z^k}{k!} = e^z$

so where $z = i x$

$\lim_{n \to \infty}(1 + \frac{i x}{n})^n = e^{i x} = \cos_x + i \sin_x$

Thepigdog (talk) 12:04, 2 March 2014 (UTC)

What you're proposing is almost the same as the limit definition based proof in the article. In that proof there is a parenthetical "(The proof of this is based on the rules of trigonometry and complex-number algebra.[6])" which is actually referring to the arguments that (1) Multiplying by (cos theta + i sin theta) corresponds to a rotation by angle theta in the complex plane; (2) In this context we can correctly approximate (1 + i*a) with (cos a + i sin a) for small a. You might say, why not actually make those arguments, like you're proposing here? Why skip them via the parenthetical? It's because we want brevity and clarity, and also we want to make it accessible to people who are unfamiliar with limits and calculus, and also because the animation helps fill in those gaps. (In my opinion anyway.) --Steve (talk) 05:08, 5 March 2014 (UTC)
Yes I am not disagreeing with you. I like the limit definition based proof section. I just think that this is the most intuitive fundamental proof. It actually describes what is happening. The other proofs (calculus and taylor series) pull a rabbit out of the hat and say "look its equal". But I am not at all happy with the above yet. Actually I am still fiddling with trying to make it more rigorous and more understandable. Maybe it will come to nothing. Hope this is OK.
Thepigdog (talk) 05:30, 5 March 2014 (UTC)
The proof is kind of the way I would like. To me the limit definition section is not really a proof. But its a very nice demonstration. I suggest it should go earlier, and be replaced by the above proof. Maybe. What do you think?
Thepigdog (talk) 11:43, 5 March 2014 (UTC)
Why is the limit definition section "not really a proof"? Because it leaves out steps? (Yours leaves out steps too...)
I don't see how your proof is "intuitive", I think most non-expert readers will see it as a series of random formal manipulations that somehow magically arrives at the right answer. (Especially readers who don't have much experience with or understanding of limits.) But maybe you're going to add more prose? --Steve (talk) 13:27, 5 March 2014 (UTC)
For example, cos x + i sin x = (cos x/n + i sin x/n)^n. In the limit of large n, cos x/n approaches 1 and i sin x/n goes to zero, so cos x/n + i sin x/n approaches 1. Therefore cos x + i sin x = 1^n = 1. Oops!
My proof is wrong, but what's wrong with it? It gives the wrong result, but that's just the symptom, not the cause. I'm sure you can answer the question, but could a typical reader read through your proof and understand why it is correct to replace sin d with d for small d, but it's incorrect to replace sin d with 0 for small d? After all, both limits are correct.
There's nothing wrong with leaving out essential details if you say that you're leaving out essential details, but if you present a certain step as a simple logical step ("replace sin d by d for small d") when it's actually fraught with danger and only works in certain circumstances, it's a problem.
Here's an example of "intuitive". The De Moivres formula (cos x + i sin x)^n = cos nx + i sin nx is not intrinsically "intuitive" -- when someone looks at the formula they won't "see" anything but a mess of symbols, unless they know what to look for. Whereas an expert might see it as quite intuitive: "The left hand side is rotating the complex plane by x radians, n times, and the right hand side is rotating the complex plane by nx radians all at once. Obviously those transforrmations are the same." In other words the De Moivre formula is a distraction from a more important and also more intuitive fact: "Multiplying by cos x + i sin x corresponds to rotating a point in the complex plane by x radians." Well it's not intuitive at first, nothing is, but it is a better building block for constructing the mental model in which Euler's formula is intuitive, better than writing down De Moivres formula as a starting point.
Anyway, it's good that you have built mental models in which statements like "sin d approaches d for small d" are intuitive, but don't forget that there will be many readers for whom facts like these statements are not intuitive... --Steve (talk) 16:03, 5 March 2014 (UTC)
Firstly I must apologise for my approach. I started this as a bit of a doodle to see if I could construct something that was the way I could understand it. I never expected to change the actual text.
I will start by saying that I really like what the author is trying to do in the limit definition section. I think in an encyclopedia its just as important to be understandable as accurate.
Yes its not wrong to leave out stuff that is too technical.
However for me I wanted a series of logical steps that took me from de moivras to e raised to the power of i x, along the way explaining what raising to the power of a complex number is. I am thinking more about how my self as back when I was a 14 year old would have understood it.
That 14 year old would have liked your section but found it frustrating also.
Yes there are still gaps in my version. Bits where the actual mechanics are not completely explained. Maybe that can be improved.
For me personally I see the limit definition as a very valuable explanation before the proof.
You also make the point that we dont really need to start with de moivras formula, which is true. in fact in an earlier version I didnt. But as it evolved I found that seemed to be the best starting point, unless I wanted to explain the whole complex number / trigonometry / field relationship.
Yes the replacing sin d with d step bothers me. I wish I had a better way of explaining that.
Well I am not going to change anything in the article unless I have agreement. I don't feel qualified to do that.
Thepigdog (talk) 21:42, 5 March 2014 (UTC)
I am really not happy with the "d = sin d" step. Its not really a logical step, and it doesnt really explain why the substitution is valid. I've never been really happy with it. I will see what can be done with it.
Thepigdog (talk) 00:38, 6 March 2014 (UTC)
I have sorted out that step (I think).
My version of the proof assumes a knowledge of limits, while yours explains the limit idea as part of the proof.
Your "imaginary part" graph is also very intuitive. To me they are too alternate approaches to essentially the same proof, but aimed at different audiences. I know you don't want any change ;). Lets consider the matter.
Thepigdog (talk) 05:03, 6 March 2014 (UTC)
Reading the section "Using the limit definition" I felt there was something missing. I suggest the above as a sub section with a title "In detail". But I don't feel that strongly about it. The argument for it is that it provides a step by step proof. On the negative side it adds to the length of the article.
Whats your thoughts on this?
Thepigdog (talk) 05:57, 6 March 2014 (UTC)
For one thing, "e^z is by definition equal to the limit of (1+z/n)^n as n --> infinity" is a legitimate definition of e^z, described as a valid definition in the article and in many references. It is just as valid and common a definition as the definition you prefer, "e^z is by definition equal to 1 + z + z^2/2 + z^3/6 + ...". So you can delete the whole half of the proof where you go from one to the other.
Well, maybe it's worth having a proof somewhere on wikipedia that the limit of (1+z/n)^n is equal to 1 + z + z^2/2 + .... In this article, it would go in the definitions section, not the proof section, as it is a proof that the two definitions offered here are actually equivalent. Or it could be in the article exponential function (and this article could just have a link), or somewhere else, I don't know.
Once you split that part off, you will see that your proof is getting more and more similar to the "limit definition" section. Actually the main difference is that that section goes forward from (1+ix/n)^n to (cos x + i sin x), while your proof goes backwards from (cos x + i sin x) to (1+ix/n)^n. Also, you are giving more detail, especially the step "(The proof of this is based on the rules of trigonometry and complex-number algebra.[6])".
Actually there used to be more detail in the proof -- see [1]. It was shortened on request to improve the readability of the article as a whole, and because interested readers can look at the cited reference to get details...
Oh, and sorry if I'm biased :-D --Steve (talk) 15:14, 6 March 2014 (UTC)
For the general puplic e^z for complex z is the natural extension you get if you extend e^y to complex numbers. Both powers series and the limit fit this requirement, and there equal. Yes this could be covered under exp but its not. I suspect if I tried to add it the exp article the mathematical mafia would stop me ;)
It seems to me that bit is of interest particularly for eulers formula. Logically you are right it should go in the exp article but practically I would want it in this article. I would be happy if it was in a separate section down the bottom of this article.
The rest of the proof is simple and in some ways easier to understand than the existing limit proof. Not that I don't like the existing limit proof, its just a different style for a different audience.
Yes that old version uses the absolute value/arg approach. Thats nice, but for me not as direct a method. Actually I prefer it to the current version. Now I see where your current version comes from. But its a different approach from the above. Its a very nice version of the absolute value, arg approach, better than other versions I have seen.
I think most people dont like to follow cited references. You never know what you might find there. It might be too detailed and assume a level of knowledge you don't have.
Cant we have both versions? And the old version too. Different levels of detail for different people.
Personally I want,
• An introductory section with your cut down proof similar to the current Using the limit definition
• In the proof section;
• The old absolute value arg proof .
• The above de moivre based proof (as above without the table).
• At the bottom of the document before references
• Proof the limit verion and taylor series are equal and are the most natural extension of exp for real to complex numbers (the table in the above).
Aaaaaaaggghhh, I love having proofs in articles, just not long detailed ones. :-(
I can see now the mathematical mafia will not allow a 4th proof, even though its the simplest :-(
Thepigdog (talk) 22:18, 6 March 2014 (UTC)
Dear mathematical mafia.
Its not the length of the dance that counts, or the number of movements. Its how beautiful the dance is and how it enlightens the world.
Firstly we don't have more proofs because more proofs makes it more true. We have more proofs because each proof tells a different aspect of why it is true. You could have 5 or more proofs and each one would give a different insight as to why this beautiful result is true,
• From De moivre formula - Very simple and highlights the role or rotations. The most elemental and constructive proof.
• Arg / absolute value - The distance from the center is one. The angle changes under rotation. Easy to visualise.
• The taylor series proof - The practical efficient proof. The exponential function of complex number must be understood as a power series. So jump ahead and use this to give an elegant powerful proof that tells you nothing about why it is true, but magically gives you the answer.
• Calculus proofs - The derivatives of exponential and trigonometric functions are known. Use them to prove a result you already know is true.
• Arg - log proof - Logarithms and angles in polar co-ordinates behave the same way. So no need to take limits. Characterize and compare there behavior to give the result.
Thats 5 fundamentally different proofs of eulers formula and each one gives insight.
Secondly proofs or derivations may be pitched at different levels of understanding. A demonstration pitches the idea of the proof for the audience not yet ready to understand the full story.
A derivation builds an explanation of core concepts (like limits) into the proof and explains them along the way.
A proof may have gaps in an encyclopedia, because not all steps are interesting. But ultimately you don't make a subject more understandable by missing out key steps. You give the illusion of understanding without true understanding. True accessibility comes from organizing the information so the readers sees, at each stage, what they need to move to the next step.
Grumble grumble grumble.
Thepigdog (talk) 13:48, 7 March 2014 (UTC)
Doggie, my whole problem with this is that the only method I know of to show that De Moivre's formula is true is by way of Euler's. Then to use that to prove De Moivre's appears ostensibly circular. We had two very simple proofs that made no assumptions other than i 2=-1, one based on a simple 1st-order derivative, the other based on a simple 2nd-order diff eq. Both of those proofs, which were nothing but solid and instructive, were removed by the mathematical mafia. So don't feel alone, but I sure don't get your "proof" when I don't understand how to prove De Moivre without first knowing about Euler. 70.109.183.141 (talk) 14:46, 8 March 2014 (UTC)
70.109.183.141: The article does have a proof based on the 1st order derivative of e^ix -- [2]. What is the "removed by the mathematical mafia" proof based on the 1st order derivative that you're talking about? I'm sorry I don't remember.
I'm not the boss of this page, and I'm not necessarily opposed to more proofs, or different proofs. I was just voicing my initial opinions and asking questions and having a discussion. :-D
A basic problem is that this article covers more than just the proof; so if the proof section is too long, the page becomes unbalanced and hard-to-read. Perhaps there should be a spin-off article Proof of Euler's formula. --Steve (talk) 00:41, 10 March 2014 (UTC)

As I think you may have discovered by looking at the talk page archives, there has been ALOT of argument in the past about what proofs to include here (some of which I took part in). Everyone seems to have a strong opinion about which proof is best, and some people have strong opinions about which shouldn't be included. My favorite has always been a variant of the calculus proof. There was an old version of the page with a better incarnation of the calculus proof (IMHO) which got destroyed by the MM (mathematical mafia). Personally I think the current limit proof is not a proof at all as it's written. It basically says, look, if you put 1000 in for n that's really close to what it should be therefore, proof done (and here's an animation in case you're not convinced). That doesn't mean I don't think it should be in the article; it is a sketch of a proof. At any rate, I would support you changing that one to be more rigorous. Holmansf (talk) 11:43, 13 March 2014 (UTC)

The limit section is not a proof, it's a summary of a proof. Isn't that the case for all of them?
I used 1000 instead of "n" because over time, three different people had made edits to the proof that made it quite clear that they couldn't follow it. Maybe you forget what it was like when you were first algebra many years ago, but any schoolteacher will tell you that most people have an easier time with numbers than with variables, especially several variables at once. You are welcome to switch it back, but you should know that it will not actually make the arguments any more or less rigorous.
I mean, saying "let's use 1000 as an example, but these statements become more and more accurate if you replace 1000 with larger and larger numbers" is exactly the same as saying "let's use n, and take the limit as n goes to infinity". Maybe the second one sounds more professional, and certainly the second one is easier for a professional mathematician to read, but I do think that the first one is easier for a budding mathematician to understand. Just my opinion, I don't really know!
You should go through the page history to find the calculus proof you prefer. I'm open-minded (more open-minded than you might think, given my previous posts). --Steve (talk) 15:53, 13 March 2014 (UTC)
No, I think the limit one is substantially farther from being a full proof than the other two which are essentially complete. Both the others contain all essential steps in some form (well the calculus one is missing a justification that the derivative of $e ^{ix}$ is $i e ^{ix}$; if you look at the comments above I argued quite hard to include a few lines explaining this but was thwarted by the MM). The limit proof is just a description of what is meant by the word "limit", and then a parenthetical hand wave: (The proof of this is based on the rules of trigonometry and complex-number algebra.) The entire meat of the proof is dismissed in that statement. Holmansf (talk) 10:06, 14 March 2014 (UTC)
This is the old version of the calculus proof which I prefer. If no one shows up to complain about it in the next few days perhaps I will put it back in.

"Several other proofs are based on the following identity obtained by differentiating the power series definition of eix. Indeed, since this series converges absolutely for all complex numbers we can differentiate it term by term to obtain

\begin{align} \frac{\mathrm{d}}{\mathrm{d}x} e^{ix} & = \frac{\mathrm{d}}{\mathrm{d}x} \sum_{n=0}^{\infty} \frac{(ix)^n}{n!} \\ & = i\sum_{n=1}^\infty i^{n-1} \frac{x^{n-1}}{(n-1)!} \\ & = i e^{ix}. \end{align}

Now we define the function

\begin{align} f(x) = (\cos x - i \sin x) \cdot e^{ix}. \end{align}

The derivative of ƒ(x) according to the product rule (note that the product rule can be proved to hold for complex valued functions of a real variable using precisely the same proof as in the real case) is:

\begin{align} \frac{d}{dx}f(x) &{}= (\cos x - i\sin x)\cdot\frac{d}{dx}e^{ix} + \frac{d}{dx}(\cos x - i\sin x)\cdot e^{ix} \\ &{}= (\cos x - i\sin x)(i e^{ix}) + (-\sin x - i\cos x)\cdot e^{ix} \\ &{}= (i\cos x + \sin x - \sin x - i\cos x)\cdot e^{ix} \\ &{}= 0. \end{align}

Therefore, ƒ(x) must be a constant function in x. Because ƒ(0)=1 in fact ƒ(x) = 1 for all x , and so multiplying by cos x + i sin x, we get

$e^{ix} \ = \cos x + i \sin x.$"

Holmansf (talk) 10:31, 14 March 2014 (UTC)