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== A simple explanation/proof for Euler's formula ==

This is technically original research, but on the other hand anyone can show that it is true. Everything about e^ix = cos x + i sin x can be understood by thinking about i^x.

you can identify a point on a unit radius circle by saying by how much 1 + 0i would have to be rotated to get there.

you can rotate a point on the Complex plane by multiplying it by powers of i, or in other words, by i^x.

therefore you can identify a point on a unit radius circle by saying by how much 1 + 0i would have to be multiplied by i^x to get there.

therefore you can identify a point on a unit radius circle in terms of just i^x.

a multiplication by i rotates a point by quarter of a circle. Therefore, you can say that any point indicated by i^x can also be indicated by cos x + i sin x, where x, cosine and sine are working in a system that divides a circle up into 4 angles.

you can rephrase i^x so that x can be a value in degrees or radians or any other way of dividing up a circle: e.g. i^(x/90) works with x in degrees; i^(2x/pi) works with x in radians. It will still be the case that any of these new exponentials will still be equal to cos x + i sin x, where x, cosine and sine are working in that particular way of dividing up a circle.

you can rephrase those exponentials to be a base raised to an Imaginary power e.g. (The 180i root of -1)^ix works in degrees; (The i*pi root of -1)^ix works in radians. It's still the case that these are equal to cos x + i sin x, where x, cosine and sine are working in that particular way of dividing up a circle

you can swap these bases for Real numbers (by using knowledge of how to calculate e^i) to get a Real number raised to an Imaginary power. So, 1.01761^ix works for degrees; e^ix works for radians. And it is still the case that these will be equal to cos x + i sin x, where cosine and sine are working in the particular way of dividing up a circle. In other words: 1.01761^ix = cos x + i sin x, when x, cosine and sine are working in degrees; e^ix = cos x + i sin x, when x, cosine and sine are working in radians.

I explain it in depth here: http://www.wimtarriner.com/

Even if this understandably gets dismissed as original research, I think the article should point out that e^ix = cos x + i sin x only works in radians. You often see it used with x in degrees, which is wrong. [[User:Timtimw|Timtimw]] ([[User talk:Timtimw|talk]]) 12:15, 6 January 2019 (UTC)
: This is not only original research, but also this contains many errors:
:* This would require a definition of <math>i^x</math> for non-integer {{mvar|x}}. The simplest definition passes by the formula <math>i^x=e^{x\log i},</math> which, in turn requires the definition of log ''i''. The common definition for this uses Euler's formula. So your whole reasoning is essentially circular.
:* Euler's formula is an equality between complex valued functions of a real variable. No measure unit is involved. So your edit request, at the end is nonsensical, as well as all comments about how measuring angles.
:* Your reasoning is sketchy on the most difficult part, the definition of [[exponentiation#Powers of complex numbers|exponentiation with a complex basis]]. It misses the fact that if {{mvar|x}} is not a [[rational number]] <math>i^x</math> has infinitely many values, and these values are dense on the unit circle; that is, for every [[irrational number|irrational]] {{mvar|x}}, and every complex number {{mvar|z}} of modulus 1 (that is lying on the unit circle), there are values of <math>i^x</math> that are as close as one want from {{mvar|z}}.
:These are only the most evident errors. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 14:40, 6 January 2019 (UTC)

:::Thanks for your contribution!

:::i^x where x is not an integer is perfectly valid. The most obvious examples you will know are i^0.5 and i^pi.

:::i^x doesn't require knowledge of e^ix to be calculated -- You can calculate simple values of x using a compass, ruler, protractor and a bit of thought.

:::Multiplication of a point on the complex plane by e^ix rotates that point by x radians. You can test this is true by picking a complex number, plotting it on axes, drawing a circle centred on the axes with a circumference that goes through that point, then drawing a second point one radian around that is still on the circumference. Its position will be the original point multiplied by e^1i. You can try this with any number of radians too. If you multiply 1 + 0i by e^1i, you will get to a point that is at cos x + i sin x, when x, sine and cosine are in radians. If e^1i rotates by 1 radian, then it cannot be the case that e^1i will rotate by 1 degree. Therefore, e^ix = cos x + i sin x cannot make sense if x is anything but radians. If you want an exponential that rotates by degrees you need to use 1.017606491206^ix. [I'm leaving in the unnecessary 1s and 0s in the complex numbers to make this clearer]. [[User:Timtimw|Timtimw]] ([[User talk:Timtimw|talk]]) 09:45, 8 January 2019 (UTC)
::::[[User:Timtimw|Timtimw]], you are missing several mathematical fundamentals that are required to discuss this in a [[mathematical rigor|rigorous]] sense. Your explanations consist purely of [[hand-waving]]. Using a visual example (or even any finite number of them) cannot by its nature constitute a [[mathematical proof]] of Euler's formula; there is no such thing as a "proof by example".
::::#The [[complex logarithm]] is a [[multivalued function]] and therefore one cannot talk about "i<sup>x</sup>" without ambiguity, unless one first specifies a particular [[branch cut]]. You fail to provide a rigorous definition of that function. In fact, all possible values of <math>i^\pi</math> ''cannot'' be constructed by compass and straightedge from the line segment from the origin to 1.
::::#The very geometric interpretation (rotating) that you talk about is only possible in the ''first place'' with Euler's formula already established.
::::#To answer your original post, sine and cosine are understood to use radians in an analytical context like this one. It may indeed be useful to clarify that, but if so, only one single sentence, because the vast majority of readers will have been exposed to radians.
::::I appreciate your curiosity in this subject, to be clear, but it should also be noted that this is [[WP:NOTAFORUM]] for discussing the mathematics of your suggestion beyond why it is unsuitable for the article. Further questions should be directed to [[Wikipedia:Reference desk/Mathematics]].--[[User:Jasper Deng|Jasper Deng]] [[User talk:Jasper Deng|(talk)]] 10:07, 8 January 2019 (UTC)

Is there an elementary way to establish at least the logarithmic form of this formula, starting from [[de Moivre's formula]]?--[[Special:Contributions/109.166.134.178|109.166.134.178]] ([[User talk:109.166.134.178|talk]]) 18:00, 22 April 2019 (UTC)
:IMO, the simplest proof of Euler's formula is to remark that both <math>y_1=e^{ix}</math> and <math>y_2=\cos x+ i\sin x</math> are solutions of the differential equation <math>y'=iy</math> and are equal for {{math|1=''x'' = 0}}. This uses a theorem of uniqueness of the solutions of differential equations, but, here, the proof is elementary, as the differential equation and the rules of differentiation show that <math>\frac{d}{dx}(y_2/y_1)=0,</math> and thus that <math>y_2/y_1</math>  is a constant, equal to 1, its value for {{math|1=''x'' = 0}}. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 21:50, 22 April 2019 (UTC)

== Simpler Proof(s) by Differential Equations ==

There are two exceedingly simple proofs of Euler's Formula, and I'm not sure why they aren't listed. Unfortunately, I don't know how to write math in HTML or TeX, so this may be a little hard to read (sorry!). So the first method is to just differentiate (cos(x) + isin(x)) / e^ix, yielding zero. Thus the function is constant, and that constant can be found to be 1 by plugging in 0 for x. Therefore the top and bottom are equal. The other way is to differentiate y = cos(x) + isin(x), yielding dy/dx = iy. Then divide both sides by y, integrate, eliminate the constant by initial value y(0) = 1, and then exponentiate, and there you are; e^ix = y = cos(x) + isin(x). Both of these are far faster and easier than the differential equation method listed in the article. It feels like one or both should be listed instead. Am I missing something? [[User:Cpotisch|Cpotisch]] ([[User talk:Cpotisch|talk]]) 04:09, 31 August 2020 (UTC)
:The proof linked to in the section "Using differential equations" is exactly your first proof. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 07:28, 31 August 2020 (UTC)

::{{reply to|D.Lazard}} Whoops. I misread the title of the polar coordinates proof as being a differential equations proof, so that’s the one I was trying to reference. But actually, is that one even valid? Doesn’t it assume that e^ix is a well defined complex number, when it isn’t necessarily?

::Also, the method linked to in the trig article is essentially an amalgam of the two I suggested, and I think it‘s presented as being more complicated than it is.

::So is there any reason not to include one of these proofs in the article? [[User:Cpotisch|Cpotisch]] ([[User talk:Cpotisch|talk]]) 15:31, 31 August 2020 (UTC)

== about [[Integration using Euler's formula]] ==

It seems better to merge Integration using Euler's formula into this page or move to wikibooks.--[[User:SilverMatsu|SilverMatsu]] ([[User talk:SilverMatsu|talk]]) 15:18, 4 September 2021 (UTC)
:'''Oppose'''. If a reader is interested by this subject, it is probably because they want to integrate a trigonometric function, not because they are searching for applications of Euler's formula. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 16:17, 4 September 2021 (UTC)
:'''Comment'''. Apparently an article is lacking, which lists the classes of functions for which there is an algorithm for computing the antiderivative. This method of integration could appear there for the integration of rational functions of trigonometric functions (as an alternative of half angle substitution), and for the integration of polynomials in {{mvar|x}}, <math>\cos x,</math> and <math>\sin x</math> (followed by integrations by parts; for this class of functions, I don't know any alternative). [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 16:17, 4 September 2021 (UTC)
::The wikibooks seem to have the following page; [[wikibooks:Calculus/Integration techniques]]--[[User:SilverMatsu|SilverMatsu]] ([[User talk:SilverMatsu|talk]]) 16:55, 4 September 2021 (UTC)

== Remove Three-dimensional visualization ==

I propose to remove the picture from the article for the following reasons
* the picture is not referenced in the text
* the picture has German labels
* the meaning of the inset scale increasing from zero to 4pi is not obvious
* use of <math>z</math> for a real angle 
* use of j for the imaginary unit in contrast to the notation used in the article.
* the picture doe not add anything to the information already depicted in the other figures.
[[Special:Contributions/141.89.116.54|141.89.116.54]] ([[User talk:141.89.116.54|talk]]) 12:38, 12 November 2021 (UTC)
: I add
:* The picture is not understandable for most readers of this article.
: I have removed the picture. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 12:54, 12 November 2021 (UTC)

== Relation to trigonometry ==

[[Euler's formula#Use of the formula to define the logarithm of complex numbers]] contains the rather weak statement 
<blockquote>Finally, the other exponential law
<math display="block">\left(e^a\right)^k = e^{a k},</math>
which can be seen to hold for all integers {{mvar|k}}, together with Euler's formula, implies '''several''' [[trigonometric identities]], as well as [[de Moivre's formula]].<br>(Emphasis added.)</blockquote> 
I believe that [[Euler's formula#Relationship to trigonometry|#Relationship to trigonometry]] should contain a much stronger; while http://mason.gmu.edu/~smetz3/humor/Euler.pdf is intended as a humorous T-shirt, it was inspired by my HS Trigonometry class, in which I never bothered to memorize the identities, but just worked them out as I needed. Perhaps
<blockquote>The definitions of the trigonometric functions and the standard identities for exponentials, together with Euler's formula, are sufficient to easily derive most trigonometric identities.</blockquote>
would be an appropriate addition. Or is that TMI? --[[User:Chatul|Shmuel (Seymour J.) Metz Username:Chatul]] ([[User talk:Chatul|talk]]) 13:34, 22 June 2022 (UTC)

:I like it.--[[User:Bob K|Bob K]] ([[User talk:Bob K|talk]]) 02:36, 23 June 2022 (UTC)

== Is that theta or x? ==

'x' is used instead of 'theta' in many parts of the article. Isn't the use of 'theta' more common? [[User:Bera678|Bera678]] ([[User talk:Bera678|talk]]) 14:32, 23 December 2023 (UTC)

:Please, do not introduce incoherencies as you did, by changing "x" into "theta" in a formula, and keeping "x" in the beginning of the sentence and in the end of the paragraph. This said, "theta" is possibly more common in pure trigonometry, that is when the variable represents explicitly an angle. "Phi" is also commonly used in some contexts, such as in electrical engineering (and in the  infobox of the article). But "x" is more common in [[calculus]] and [[mathematical analysis]], where it needs not representing an angle. However, in all cases, any letter would be mathematically correct. 
:It is thus correct to use "x" everywhere, except when talking of polar coordinates. So the whole article uses coherent notation except the infobox and the section {{alink|Using differentiation}}, where "x" would be more coherent than "theta". However, there is no real harm to leave this section as it is. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 16:14, 23 December 2023 (UTC)
::OK [[User:Bera678|Bera678]] ([[User talk:Bera678|talk]]) 17:59, 23 December 2023 (UTC)

== False rearrangement in series proof ==

I don't think that the line "The rearrangement of terms is justified because each series is absolutely convergent" is great here because the above manipulations aren't rearrangements at all. A rearrangement would entail just one infinite series. What we have can't be written as a relabeling based on a permutation of the naturals.

Although this splitting of a series into the series of its odd and even terms certainly feels like a rearrangement, it's really just bracketing. This bracketing is justified because the original series is absolutely convergent, and the result relies on this and that the two subseries are each convergent (they needn't be absolutely convergent).

I think that it would be more appropriate to say "The splitting of terms is justified because each series is convergent, and the original series is absolutely convergent." [[User:OisinDavey|OisinDavey]] ([[User talk:OisinDavey|talk]]) 20:41, 18 February 2024 (UTC)

== Translation, please. ==

Could someone write a translation of the article in a way that us "C" students can understand? [[User:Nosehair2200|Nosehair2200]] ([[User talk:Nosehair2200|talk]]) 20:54, 21 August 2024 (UTC)

:It would be helpful if you could pick one item that was particularly confusing.  There are not many people here who have the time to rewrite the article but there are a few that could polish one small part. [[User:Constant314|<b style="color: #4400bb;">''Constant314''</b>]] ([[User talk:Constant314|talk]]) 21:21, 21 August 2024 (UTC)

== Variable z is in the wrong place for the limit formula ==

With f(z)=e^z=lim[n->infinity] you now have (1+z/n)^n but this should be (1+1/n)^(n*z) and the fact that it isn't becomes really clear when you do the math for the derivative.  <!-- Template:Unsigned --><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Emilehobo|Emilehobo]] ([[User talk:Emilehobo#top|talk]] • [[Special:Contributions/Emilehobo|contribs]]) 15:28, 23 September 2024 (UTC)</small> <!--Autosigned by SineBot-->

:Your formula is correct only for real {{mvar|z}}. It cannot be used for defining the exponential function, since it contains a an exponent that is not an integer, and the exponentiation with non-integer exponents requires the exponential function for being defined. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 16:22, 23 September 2024 (UTC)
::Okay. I figured I needed to do a bit of math with my calculator for backup for the real numbers of 1 and -1 that are also a part of the formula, but it does check out was what I thought, but then I realized z doesn't equal 1 or -1, so how do you handle that? Isn't it written wrong?
::I'm worried about the conflicting definitions a bit and think it might benefit clarification to contrast the two views of the use of e also in the limit definition. The plain English guy up above has a point.
::One difficulty I don't see the answer for yet is whether you treat "infinity" as an even or an odd number, which probably has major implications also.
::Where do I find how you handle (e^(i*psi))^k also, because now it just says it multiplies your angular momentum with k, but how do you make sure people reading the work instantly recognize it and don't mistake it for regular e^k or e^-k? The limit function doesn't seem to differentiate and I think it might have implications.
::The main problem I think is that the exact same thing on paper can now mean two things, so maybe differentiate between the two by calling one "e" and the other "e_psi" also? (Underscore = subscript.)
::Out of curiosity, for determining e, has anyone ever tried: SUM[forall p=prime]:p^-1 ? [[User:Emilehobo|Emilehobo]] ([[User talk:Emilehobo|talk]]) 20:05, 23 September 2024 (UTC)
:You can think of the exponential as a kind of limit of power functions. A power function {{tmath|z \mapsto z^k}} for any arbitrary {{tmath|n}} wraps the complex plane around the origin, conformally mapping the flat plane onto a cone with a singularity at the origin (the vertex of the cone). If {{tmath|n}} is an integer, then the the unit circle gets completely wrapped around {{tmath|n}} times. To make things more consistent from (a portion of) one cone to another, we can offset the points we examine to the multiplicative identity 1, and divide by {{tmath|n}} to normalize the scale at that point. The exponential function is what you get when you take the limit as the cone becomes infinitely pointy, turning it into a cylinder. This moves the origin vertex away to infinity, but we still have nice behavior near 1. –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 17:58, 23 September 2024 (UTC)
::I'm not saying you're right or wrong, but this is an encyclopedia... There are also a lot of conflicting statements in what you say, which is really cool in terms of metaphores that you are now using, but for instance a limit limits and exponentials for numbers greater than one very quickly want to go to infinity without halt... But I do seriously love someone explaining math like Jim Morrison would. You are definitely beautifully wicked. [[User:Emilehobo|Emilehobo]] ([[User talk:Emilehobo|talk]]) 21:08, 23 September 2024 (UTC)

== Circular reasoning in proof by differentiation ==

In the first proof listed, namely the one using differentiation, there might be a circular reasoning. Both the sine and cosine functions can easily be proven to be differentiable by using the Cauchy-Riemann equations, but although I have been searching a lot, I can't seem to find a proof of the differentiability of the natural exponential function for complex numbers without using Euler's formula. Do anyone know such a proof? [[User:Tøger Holm Lyngbo|Tøger Holm Lyngbo]] ([[User talk:Tøger Holm Lyngbo|talk]]) 12:45, 14 December 2024 (UTC)

:You can see {{slink|Exponential function#Derivatives and differential equations}}: all given definitions and their equivalence proof extend verbatim to the complex case, excepted the definition as inverse of the logarithm. Also, the definition through power series implies complex differentiability as it is the case for every [[analytic function]]. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 16:05, 14 December 2024 (UTC)

== about hatnote ==

Edit [[special:permalink/1280169638|permalink/1280169638]], reverted by [[special:permalink/1280172228|permalink/1280172228]], added a bunch of text about [[Euler's polyhedron formula]]. The {{tl|about}} template links to {{slink|List of things named after Leonhard Euler|Formulas}}, which includes {{tqq|[[Euler's polyhedral formula]] for planar graphs or polyhedra: {{math|1=''v'' − ''e'' + ''f'' = 2}}, a special case of the [[Euler characteristic]] in topology}}. Is it important enough to warrant including it in the hatnote, or is the existing link enough? -- [[User:Chatul|Shmuel (Seymour J.) Metz Username:Chatul]] ([[User talk:Chatul|talk]]) 13:41, 13 March 2025 (UTC)

:Isn't the polyhedra formula a totally unrelated formula to the formula discussed in this article? [[User:Constant314|<b style="color: #4400bb;">''Constant314''</b>]] ([[User talk:Constant314|talk]]) 14:08, 13 March 2025 (UTC)
::It seemss that {{u|Chatul}} does not challenge the revert. Instead they ask the question and whether the formula on polyhedra deserves a special mention in the hatnote. My opinion that the answer must be '''no''', for two reasons: firstly, the formula is better known as "Euler's characteristic formula" or simply "[[Euler characteristic]]". Secondly, giving a special importance to Euler characteristic, requires a consensus; otherwise, this would be [[WP:Original synthesis]]. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 14:42, 13 March 2025 (UTC)
:::So, we already have an article on the polyhedra formula.  If the reverted material has a reliable source, it could go there. [[User:Constant314|<b style="color: #4400bb;">''Constant314''</b>]] ([[User talk:Constant314|talk]]) 14:57, 13 March 2025 (UTC)
:::Correct. I've usually seen it referred to as ''Euler's formula'', but it has a variety of names. -- [[User:Chatul|Shmuel (Seymour J.) Metz Username:Chatul]] ([[User talk:Chatul|talk]]) 15:06, 13 March 2025 (UTC)
::::If you think it is important enough to add it as an explicit item in the existing hatnote, that would be fine with me. [[User:Constant314|<b style="color: #4400bb;">''Constant314''</b>]] ([[User talk:Constant314|talk]]) 17:52, 13 March 2025 (UTC)
:::I don't think hatnotes are original research (they are navigational aides, not encyclopedia content), but it seems unnecessary to call this out specifically. There are a lot of Euler "formulas". –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 22:38, 16 March 2025 (UTC)
::::Well, Euler's [[Seven Bridges of Königsberg]] problem and polyhedral formula are arguably the originas of [[algebraic topology]]/ -- [[User:Chatul|Shmuel (Seymour J.) Metz Username:Chatul]] ([[User talk:Chatul|talk]]) 13:06, 17 March 2025 (UTC)