Talk:Exponential function

Parentheses
Sometimes there's exp(z), sometimes there's exp z. Shouldn't an article be consistent with itself? — Preceding unsigned comment added by A1E6 (talk • contribs) 17:57, 2 July 2019 (UTC)


 * For the most part, yes. MOS:MATH 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 $$\exp(x+y),$$ but not in $$\exp x,$$ or even something like $$\exp ix.$$  –Deacon Vorbis (carbon &bull; videos) 19:41, 2 July 2019 (UTC)

Extinction coefficient?
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)


 * ✅ I have documented this in this article and also in Exponential decay.—Anita5192 (talk) 19:47, 21 May 2020 (UTC)

Correction to my edit to the Overview section that got reverted (Forgot the binomial coefficients)
For example, using the definition $$\exp x = \lim_{n\to\infty}\left(1 + \frac{x}{n}\right)^{n}$$ and $$\exp y = \lim_{n\to\infty}\left(1 + \frac{y}{n}\right)^{n}$$,
 * $$\begin{align} \exp x \cdot \exp y &= \lim_{n\to\infty}\left(1 + \frac{x}{n}\right)^{n} \cdot \lim_{n\to\infty}\left(1 + \frac{y}{n}\right)^{n}\\

&= \lim_{n\to\infty}\left[\left(1 + \frac{x}{n}\right)^{n} \left(1 + \frac{y}{n}\right)^{n}\right]\\ &= \lim_{n\to\infty}\left[\left(1 + \frac{x}{n}\right) \left(1 + \frac{y}{n}\right)\right]^n\\ &= \lim_{n\to\infty}\left(1 + \frac{x+y}{n} + \frac{xy}{n^2}\right)^{n}\\ &= \lim_{n\to\infty}\sum_{k=0}^n \binom nk \left(1 + \frac{x+y}{n}\right)^{n-k}\left(\frac{xy}{n^2}\right)^k \text{(Binomial theorem)}\\ &= \lim_{n\to\infty}\left[\left(1 + \frac{x+y}{n}\right)^n + \binom n1 \left(1 + \frac{x+y}{n}\right)^{n-1}\left(\frac{xy}{n^2}\right) + \binom n2 \left(1 + \frac{x+y}{n}\right)^{n-2}\left(\frac{xy}{n^2}\right)^2 + ...\right]\\ &= \lim_{n\to\infty}\left(1 + \frac{x+y}{n}\right)^n + \lim_{n\to\infty}\left[\binom n1 \left(1 + \frac{x+y}{n}\right)^{n-1}\left(\frac{xy}{n^2}\right) + \binom n2 \left(1 + \frac{x+y}{n}\right)^{n-2}\left(\frac{xy}{n^2}\right)^2 + ...\right]\\ &= \lim_{n\to\infty}\left(1 + \frac{x+y}{n}\right)^n + \lim_{n\to\infty}\left(\frac{xy}{n^2}\right)\left[\binom n1 \left(1 + \frac{x+y}{n}\right)^{n-1} + \binom n2 \left(1 + \frac{x+y}{n}\right)^{n-2}\left(\frac{xy}{n^2}\right) + ...\right]\\ &= \exp(x + y) + 0 \text{ (as} \lim_{n\to\infty}\frac{xy}{n^2} = 0) \\ &= \exp(x + y) \\ \end{align}$$ 49.147.83.13 (talk) 16:21, 21 May 2020 (UTC) Wondering if one could improve on that for the edit to be approved
 * Again, this proof in not useful without a proof of the equivalence of the definitions. If one has the equivalence, one can use the definition through derivatives: The derivative with respect to $x$ of $$e^{x+y}/e^y$$ shows that this function is equal to its derivative, and equals 1 for $x = 0$; thus, it equals $$e^x$$ 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)
 * 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 &bull; videos) 17:05, 21 May 2020 (UTC)

The primary meaning of exponential function
In 2016 (see Talk:Exponential function/Archive 1), there was an RfC where the consensus was that the primary subject of Exponential function should be the function $$e^x$$. Given this, shouldn't the article begin with and focus on $$e^x$$ (as it did up to 2015) and define the more general variants $$ab^x$$ further down? Is it just that no one has gotten around to implementing this? Ebony Jackson (talk) 06:02, 21 February 2021 (UTC)
 * I suggest to boldly change the beginning into
 * In mathematics, the exponential function (sometimes called the natural exponential function) is the function $$f(x)=e^x,$$ where $e = 2.71828...$ is the basis of the natural logarithm. More generally, an exponential function is ...
 * and to edit accordingly the remainder of the lead. This would be an implementation of the Principle of least astonishment. Possibly the remainder of the article should also be restructured, but this is another question. D.Lazard (talk) 09:41, 21 February 2021 (UTC)

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
 * In mathematics, the exponential function, denoted $ex$, is the function that equals its own derivative and that has value $1$ at $x = 0$.  Its value at $x = 1$, denoted $e = 2.71828...$, is the base of the natural logarithm. More generally, an exponential function is ...

Ebony Jackson (talk) 16:38, 21 February 2021 (UTC)

Exponential behaviour
A function is exponential when it shows behaviour:

f(a + b) = f(a) * f(b)

Examples are "e^x", cosine and sine, selection of Taylor series.--86.83.108.100 (talk) 12:58, 29 September 2021 (UTC)


 * 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)

Branches
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)
 * 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 $$a^x=e^{x\ln a},$$ 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)

Inconsistency in the lead
In my opinion, there is an implication that all functions of the form $$ab^x$$ satisfy the identity $$f(x+y)=f(x)f(y)$$. Obviously this is not true, but I wonder then how we should introduce these exponential functions. Student298 (talk) 20:32, 6 November 2022 (UTC)

Wiki Education assignment: 4A Wikipedia Assignment
— Assignment last updated by Ahlluhn (talk) 00:57, 31 May 2024 (UTC)

Formal Definition
In the formal definition: $$ e^x \equiv \sum_{j=0}^{\infty} [x^j/j!] $$ the LHS is apparently assumed a priori (and proved later in the Overview section). Perhaps for beginners it would be more satifying not to assume this but instead prove it by considering:

(a) $$ f[x] \equiv \sum_{j=0}^{\infty} [x^j/j!] \;\;\;\;\;\; f[1] \equiv \sum_{j=0}^{\infty} [1/j!] \equiv e. $$

(b) $$ g[x] \times g[y] = g[x+y] \; \Leftrightarrow \; g[x] = B^x $$ are equivalent where B is an appropriate base.

Then show and use: $$ f[x] \times f[y] = f[x+y] \; \Leftrightarrow \; f[x] = B^x $$ by a very satisfying multiplication of this particular definition as expanded below.

Hence $$ f[1] \equiv e = B \; \to \; e^x \equiv \sum_{j=0}^{\infty} [x^j/j!]. $$


 * $$\begin{array}{l}

f[x] \times f[y]   = \sum_{j=0}^{\infty}{x^j /j! } \times \sum_{k=0}^{\infty}{y^k /k! }  \\ \\ = \left\{\begin{array}{cc} 1 	& \times (1 + y + y^2/2 + y^3 /3! + \dots) \\ +x 	& \times (1 + y + y^2/2 + y^3 /3! + \dots) \\ + x^2/2 & \times (1 + y + y^2/2 + y^3 /3! + \dots) \\ + \dots \end{array}\right. \\ \\ = \left\{\begin{array}{ccccccc} 1 	& +y/1! & +y^2/2! & +y^3/3! & +y^4/4! & + y^n/n! & +\dots \\ 	+x/1! & +xy 	& +\frac{x y^2}{1! 2!} & +\frac{x y^3}{1! 3!} & +\frac{x^{n-4}y^4}{(n-4)! 4!} & +\dots & +\dots \\+x^2/2! & +\frac{x^2 y}{2! 1!} & +\frac{x^2 y^2}{2! 2!} & +\frac{x^{n-3}y^3}{(n-3)! 3!}  & +\dots   & +\dots 	& +\dots \\+x^3/3! & +\frac{x^3 y}{3! 1!} & +\frac{ x^{n-2}y^2}{(n-2)! 2!} & +\dots     	& +\dots   		& +\dots    	& +\dots \\+x^4/4! & +\frac{x^{n-1}y}{(n-1)!1!}  & +\dots   & +\dots    			& +\dots   		& +\dots    	& +\dots \\ +x^n/n! & +\dots 	& +\dots	& +\dots 			& +\dots 		& +\dots 	& +\dots \\ +\dots 	& +\dots 		& +\dots	& +\dots 			& +\dots 		& +\dots 	& +\dots \end{array}\right. \\ \\ = \left\{\begin{array}{ccccc} 1 	         \\ 		&  +(x+y) \\ 		&	& +\frac{(x+y)^2}{2!} \\ 		&	& 	&+\frac{(x+y)^n}{n!} \\		&	&	&	& + \dots \end{array}\right. \\ \\ = \sum_{n=0}^{\infty}{(x+y)^n /n! } = f[x+y] \end{array} $$

— Preceding unsigned comment added by MikeL2468 (talk • contribs) 16:58, 26 June 2024 (UTC)


 * These huge formulas are certainly not convenient for beginners. D.Lazard (talk) 17:20, 26 June 2024 (UTC)
 * I made some changes to the article including to address this concern. Rather than giving the proofs, I merely indicate that they exist.  — Q uantling (talk &#124; contribs) 18:32, 26 June 2024 (UTC)
 * Thank you for your changes addressing this concern. I could not spot them, but I may be looking in the wrong place.
 * I wrote up these comments as it took me 60 years 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 2.2 and I think this may be related to Stirling's formula for factorials.
 * Hence 3 ways of looking at e^x - as well as the others mentioned in this article. Best wishes, Mike. MikeL2468 (talk) 19:48, 26 June 2024 (UTC)
 * 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 $k$.  — Q uantling (talk &#124; contribs) 16:52, 27 June 2024 (UTC)

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 $$b^x$$. 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)


 * 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 $f(x + y) = f(x)f(y)$.  — Q uantling (talk &#124; contribs) 17:07, 27 June 2024 (UTC)