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@A1E6: For the most part, yes. MOS:MATH#Multi-letter names touches briefly on this, but is certainly not definitive. Personally, I think having parentheses for standard functions like sin, exp, etc. tends to clutter, especially when there are parens around the whole term for one reason or another. However, having looked over quite a few math articles, my general feeling is that there are a lot (although I wouldn't automatically say most, or even a majority) of editors who include them. I'm not sure if this is because it's just their preference, or because they think it's required, or maybe for other reasons. Anyway, I'd be in favor of removing them when they're not necessary. For example, it would be necessary in an expression like but not in or even something like –Deacon Vorbis (carbon • videos) 19:41, 2 July 2019 (UTC)Reply
Latest comment: 4 years ago2 comments2 people in discussion
The equations y = e^(-kx) and y = 1-e^(-kx) are extremely common in engineering and physics, but I can't find a non-specific name for k or 1/k. In electronics 1/k is a time constant like "the RC time constant". In physics k can be called attenuation coefficient and 1/k can be the attenuation length. If you change e to 2, 1/k it is called the half-life. I think I've seen 1/k called the expected or mean life, time, or distance. Can someone think of what it is supposed to be called and include it in this article? 1/k is the "expected value" but that's too general. Ywaz (talk) 12:56, 21 May 2020 (UTC)Reply
Again, this proof in not useful without a proof of the equivalence of the definitions. If one has the equivalence, one can use the definition through derivatives: The derivative with respect to xof shows that this function is equal to its derivative, and equals 1 for x = 0; thus, it equals for every x. The proof takes only two lines and explains better why the identity is true. So, your proof is definitively not useful, except as an exercise for students. D.Lazard (talk) 16:55, 21 May 2020 (UTC)Reply
Your proof is also incorrect (although fixable) and omits a couple things even still. I leave it as an exercise to determine where the mistake is, and also as a cautionary tale about coming up with your own proofs rather than adapting ones from existing sources. –Deacon Vorbis (carbon • videos) 17:05, 21 May 2020 (UTC)Reply
Inmathematics, the exponential function (sometimes called the natural exponential function) is the function where e = 2.71828... is the basis of the natural logarithm. More generally, an exponential function is ...
Yes, that is good. Is natural exponential function common terminology? I am not sure, even though natural logarithm certainly is.
Perhaps it makes sense to think of e as an object of secondary importance, defined in terms of the important function ex, instead of the other way around? With this in mind, perhaps an alternative lead could focus more on the key property of the exponential function, as in
Inmathematics, the exponential function, denoted ex, is the function that equals its own derivative and that has value 1atx = 0. Its value at x = 1, denoted e = 2.71828..., is the base of the natural logarithm. More generally, an exponential function is ...
Cosine and sine do not satisfy that identity, you may be thinking of the complex exponential, which can be expressed in terms of cosine and sine. Furthermore, a function is exponential if it is proportional to its rate of growth, so your equation is missing a constant factor. Student298 (talk) 20:50, 6 November 2022 (UTC)Reply
Latest comment: 2 years ago2 comments2 people in discussion
I think to improve coverage the article should have some discussion about branches. Maybe the exponential function as commonly defined is only one branch, but several related functions are inescapably branched and in some contexts it's convenient to consider the exponential function as a branched function as well. The article only vaguely hints at the branching behaviour and only if you already knew what to look for when you started reading (search for ‘multivalued’). — Preceding unsigned comment added by 77.61.180.106 (talk) 01:05, 22 February 2022 (UTC)Reply
I believe that that you call "branching" is the study of multivalued functions near their singularities. As the exponential function is an entire function, there is no singularity, and no branching. This is not the case for functions when a is not real and positive. This case is considered in Exponentiation, as said in the hatnote at the top of the article. D.Lazard (talk) 10:11, 22 February 2022 (UTC)Reply
Latest comment: 1 year ago1 comment1 person in discussion
In my opinion, there is an implication that all functions of the form satisfy the identity . Obviously this is not true, but I wonder then how we should introduce these exponential functions. Student298 (talk) 20:32, 6 November 2022 (UTC)Reply
Wiki Education assignment: 4A Wikipedia Assignment
Latest comment: 1 month ago1 comment1 person in discussion
This article was the subject of a Wiki Education Foundation-supported course assignment, between 12 February 2024 and 14 June 2024. Further details are available on the course page. Student editor(s): Not Fidel (article contribs). Peer reviewers: Maaatttthhheeewww.
Latest comment: 7 days ago7 comments4 people in discussion
In the formal definition:
the form of the LHS is apparently assumed a priori (and proved later in the Overview section). For an introduction it may be simpler not to assume this but prove it from:
B is an appropriate base. Then show and use:
by a simple and symmetric multiplication of this particular definition as expanded below. Hence after all:
Another consequence of:
integer, is that the sum can be pictured in a form such that the two largest terms are:
with other terms falling off roughly as a Gaussian_function with Gaussian_integral
Although this is pulled out of a hat here, it can be reverse engineered from Stirling's_approximation for factorials. So:
where a better approximation comes by setting
Thank you for your changes addressing this concern. I wrote up these comments as it took me a long time to spot that e^x was an 'a priori' assumption.
By the way, the largest terms in e^m are the mth and (m-1)th terms, the others falling away in a Gaussian like distribution with width square-root m. Throw in a factor like 2.5 or and this is directly related to Stirling's formula for factorials.
My changes were to clarify that exp x can be defined in several equivalent ways (by power series, infinite product, or differential equation) but it then has to be proved that exp x = (exp 1)x. Also that it then has to be proved that a non-zero function f(x) satisfying f(x + y) = f(x)f(y) will necessarily be of the form exp kx for some k. —Quantling (talk | contribs) 16:52, 27 June 2024 (UTC)Reply
I think part of the problem with this article is that it's really about two different things, the natural exponential function exp and exponential functions . The article actually defines these things differently. The natural exponential is given by a series (or other equivalent characterization), whereas exponential functions are given by approximation. This schizoid nature of the article makes it very confusing. The lede is five paragraphs long, for example. To me, that's an indication that there are really two different topics here: "Elementary" exponential functions, like those of precalculus, which can be rigorously defined using only integer exponentiation, continuity, and and completeness, and the natural exponential and those derived from it. Unfortunately, there is no distinction in usage between these two topics because the "natural" exponential is strictly more general. Tito Omburo (talk) 11:27, 27 June 2024 (UTC)Reply
Wikipedia's article on exponentiation discusses expressions like bx, so I think it is right for this article to focus on exp x as defined by power series, infinite product, or differential equation. I think that the present article should mention but not go too deeply into the fact that the exponential function has an interpretation in terms of exponentiation: exp x = (exp 1)x. Likewise, I think it is appropriate to mention but not go too deeply into the fact that exp kx acts like bx and thus solves requirements like f(x + y) = f(x)f(y). —Quantling (talk | contribs) 17:07, 27 June 2024 (UTC)Reply