Talk:Euler's formula

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thread_title:  External links modified 

message: I'm confused and hoping for explanation.  How is it rationally possible to posit or envision the Cotes equation cited in this article without x being in radians and fully understood (by Cotes) as a periodic function of pi, as is stated in the article?  How did Cotes instead define x? Wikibearwithme (talk) 23:46, 13 January 2018 (UTC)
replies: text: Thanks. The article statement is unsourced and seems thoroughly dubious, and I've added a Citation Needed requesting a supporting citation or a suitable re-wording. What Cotes actually used is not too clear to me (tho I think it is basically radians as found in an arc of a quadrant of a circle, seemingly with no need to consider periodicity for the specific problem that he was dealing with, irrespective of whether he was aware of such periodicity or not - although I suppose it is conceivable that ignoring periodicity may have introduced a flaw into his proof). What he actually wrote, plus a modern interpretation thereof, can be seen in the lengthy footnote (currently numbered [5]) in his biographical article Roger Cotes (although this footnote is basically just dealing with his conclusion, and is not dealing with his proof, which is available online, but possibly only in Latin - although his archaic English terminology would presumably also be a problem for any Wikipedian trying to read it).Tlhslobus (talk) 07:05, 16 January 2018 (UTC)

text: Thanks. At the moment I'm waiting for a decent period of time (I'm never clear how long that should be) for somebody to come up with a supporting citation (which seems unlikely, but just about possible if there's an error in his proof as a result). However such waiting always carries the risk that it will never get fixed, so I won't object if anybody else decides to fix it straight away. I'm not 100% sure what the ideal rewording should be, but perhaps something along the lines of "The term Euler's formula is used because Euler's exponential formulation, although it came later, has been preferred to Cotes's logarithmic formulation because a complex number actually has an infinite number of natural logarithms due to the periodicity of trigonometric functions." Tlhslobus (talk) 06:11, 20 January 2018 (UTC)
thread_title:  Cotes equation 

message: It seems better to merge Integration using Euler's formula into this page or move to wikibooks.--SilverMatsu (talk) 15:18, 4 September 2021 (UTC)
replies: text: 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. D.Lazard (talk) 16:17, 4 September 2021 (UTC)

text: 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 , \cos x, and \sin x (followed by integrations by parts; for this class of functions, I don't know any alternative). D.Lazard (talk) 16:17, 4 September 2021 (UTC)
replies: text: The wikibooks seem to have the following page; wikibooks:Calculus/Integration techniques--SilverMatsu (talk) 16:55, 4 September 2021 (UTC)
thread_title:  about [[Integration using Euler's formula]] 

message: Euler's formula#Use of the formula to define the logarithm of complex numbers contains the rather weak statement 
Finally, the other exponential law
\left(e^a\right)^k = e^{a k},
which can be seen to hold for all integers , together with Euler's formula, implies several trigonometric identities, as well as de Moivre's formula.(Emphasis added.) 
I believe that #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
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.
would be an appropriate addition. Or is that TMI? --Shmuel (Seymour J.) Metz Username:Chatul (talk) 13:34, 22 June 2022 (UTC)
replies: text: I like it.--Bob K (talk) 02:36, 23 June 2022 (UTC)
thread_title:  Relation to trigonometry 

message: 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." OisinDavey (talk) 20:41, 18 February 2024 (UTC)
thread_title:  False rearrangement in series proof