Talk:Exponential function

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)

 Done 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 $\exp y=\lim _{n\to \infty }\left(1+{\frac {y}{n}}\right)^{n}$,

${\begin{aligned}\exp x\cdot \exp y&=\lim _{n\to \infty }\left(1+{\frac {x}{n}}\right)^{n}\cdot \lim _{n\to \infty }\left(1+{\frac {y}{n}}\right)^{n}\\&=\lim _{n\to \infty }\left[\left(1+{\frac {x}{n}}\right)^{n}\left(1+{\frac {y}{n}}\right)^{n}\right]\\&=\lim _{n\to \infty }\left[\left(1+{\frac {x}{n}}\right)\left(1+{\frac {y}{n}}\right)\right]^{n}\\&=\lim _{n\to \infty }\left(1+{\frac {x+y}{n}}+{\frac {xy}{n^{2}}}\right)^{n}\\&=\lim _{n\to \infty }\sum _{k=0}^{n}{\binom {n}{k}}\left(1+{\frac {x+y}{n}}\right)^{n-k}\left({\frac {xy}{n^{2}}}\right)^{k}{\text{(Binomial theorem)}}\\&=\lim _{n\to \infty }\left[\left(1+{\frac {x+y}{n}}\right)^{n}+{\binom {n}{1}}\left(1+{\frac {x+y}{n}}\right)^{n-1}\left({\frac {xy}{n^{2}}}\right)+{\binom {n}{2}}\left(1+{\frac {x+y}{n}}\right)^{n-2}\left({\frac {xy}{n^{2}}}\right)^{2}+...\right]\\&=\lim _{n\to \infty }\left(1+{\frac {x+y}{n}}\right)^{n}+\lim _{n\to \infty }\left[{\binom {n}{1}}\left(1+{\frac {x+y}{n}}\right)^{n-1}\left({\frac {xy}{n^{2}}}\right)+{\binom {n}{2}}\left(1+{\frac {x+y}{n}}\right)^{n-2}\left({\frac {xy}{n^{2}}}\right)^{2}+...\right]\\&=\lim _{n\to \infty }\left(1+{\frac {x+y}{n}}\right)^{n}+\lim _{n\to \infty }\left({\frac {xy}{n^{2}}}\right)\left[{\binom {n}{1}}\left(1+{\frac {x+y}{n}}\right)^{n-1}+{\binom {n}{2}}\left(1+{\frac {x+y}{n}}\right)^{n-2}\left({\frac {xy}{n^{2}}}\right)+...\right]\\&=\exp(x+y)+0{\text{ (as}}\lim _{n\to \infty }{\frac {xy}{n^{2}}}=0)\\&=\exp(x+y)\\\end{aligned}}$

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 • videos) 17:05, 21 May 2020 (UTC)

The primary meaning of exponential function

In 2016 (see Talk:Exponential function/Archive 1#Confusion in lead about what the topic is), 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 e^x, 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 e^x, 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

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.

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

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

— 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. —Quantling (talk | 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. —Quantling (talk | 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 b^x, 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 b^x and thus solves requirements like f(x + y) = f(x)f(y). —Quantling (talk | contribs) 17:07, 27 June 2024 (UTC)

exponential functions, exponentiation, exponents, power functions?

I'm an old geographic scientist with broad mathematical experience and it seems to me that Wikipedia doesn't make clear distinctions among the above concepts - any maybe some others as well. My own approach would be clear formulas and clear graphics - on the order of the thumbnail. Let me know if you'd like help with this... Otherwise, i'll keep quiet.

examples of 10 curves with real exponents between -1 and 1.25

. Lee De Cola (talk) 01:19, 20 August 2024 (UTC)

@Ldecola: Your call for 'clear distinctions' brings me to the presentation of (not very well known?) verbal characterizations of four main types of functions, together with a subtype of each.

• A lineair function transforms equidistant pairs into equidistant pairs:   f(u+s) - f(u) = f(v+s) - f(v) .

    * A proportional(?) function transforms addition into addition:   f(u+v) = f(u) + f(v) .

• An exponential function transforms equidistant pairs into equiratio pairs:   f(u+s) / f(u) = f(v+s) / f(v) .

    * An anti-logarithmic(?) function (R→R⁺) transforms addition into multiplication:   f(u+v) = f(u) • f(v) .

• An anti-exponential(?) function transforms equiratio pairs into equidistant pairs:   f(ru) - f(u) = f(rv) - f(v) .

    * A logarithmic function (R⁺→R) transforms multiplication into addition:   f(uv) = f(u) + f(v) .

• A general power(?) function transforms equiratio pairs into equiratio pairs:   f(ru) / f(u) = f(rv) / f(v) .

    * A power(?) function (R⁺→R⁺) transforms multiplications into multiplications   f(uv) = f(u) · f(v) .

A much harder point (but important for you - and for me as well) is how to get the whole WPen accept unique names in all eight cases? Short, but very artificial and therefore chanceless, should be: s-s-functions, a-a-functions, s-r-functions, etc.

About sources. It's hardly to believe that this simple scheme shouldn't be ever mentioned on WPen. Or on other WPs, or other internet-pages. Half of the scheme - the four 'subtypes' - has a famous source, 200 years old: C-A Cauchy, Cours d'analyse 1821, Chap.V, pp. 103-122. So maybe the other four types could be found in books from that period? Hesselp (talk) 18:36, 20 August 2024 (UTC)

@Ldecola Please feel free to make concrete and specific suggestions about improvements you imagine to this or related articles. We have articles Exponential function and Exponentiation (which also defines exponent); for now Power function redirects to Exponentiation § Power functions but it could certainly be its own article. All of these could, like most articles in Wikipedia, benefit from more work. –jacobolus (t) 23:18, 25 August 2024 (UTC)

Dieudonné on defining standard functions by their main property

@Ldecola: A much more recent source (compare Cauchy, above 20/08) on defining a class of functions by their collective property, can be found in  J. Dieudonné, Foundations of Modern Analysis, 1960; p.83 (4.3.7) :

"Any continuous mapping R⁺ into R⁺ such that  g(xy) = g(x) g(y)  has the form  x^a  with a real."

Four remarks:

-1. D. doesn't mention a name for the defined class of functions. Possibly(?):  "the 1 to 1 power functions".

-2. Isn't mathematical more just to use a defining condition, than to 'define' a class of functions by showing the form in which they are usually notated on paper? (In many cases several different forms are in use.)

-3. D. doesn't mention that the degree of a given power function f is equal to the, not on x dependent, value  x f'(x) / f(x) .

-4. On p.82 (4.3.2), D. uses as well the main property  f(xy) = f(x) + f(y)  to charcterize the mapping named the logarithm of base a. Hesselp (talk) 16:34, 22 September 2024 (UTC)

Dieudonné's quotation is not a definition. It is a theorem. D.Lazard (talk) 17:20, 22 September 2024 (UTC)

i'm still concerned that the articles relating to the topics i named don't help beginners get a clear idea about how exponentiation, etc is defined. the mathematical functions and operations are no doubt unambiguous, but the definitions aren't. however, i'm not qualified to clear thus up. Lee De Cola (talk) 03:02, 24 September 2024 (UTC)

Natural / general / more general exponential function(s)

@Alsosaid1987, Magyar25, and 91.170.28.20:
In the introduction the word ‘exponential’ seems to be used in four ways:
- The exponential function ($e^{x}$)
- also known as exponential functions ($b^{x}$)
- allows general exponential functions (?)
- more generally also known as exponential functions ($ce^{ax}=cb^{kx}$)
So well defined names, sometimes with alternatives, are (imho) strongly desired.
My question:  Who knows something better (at least for use in this article) than:
- General exponential function(s)   ($ab^{x}$)
- Special exponential function(s) / Exponential function(s) / Zero-to-one (f(0)=1) exponential function(s)   ($b^{x}$)
- The natural exponential function / The exponential function   ($e^{x}$)

Similar names
The nomenclature described above could be extended to:
- General logarithmic function(s)   ($a\log _{b}x$)
- Special logarithmic function(s) / Logarithmic function(s) / One-to-zero logarithmic function(s)   ($\log _{b}x$)
- The natural logarithmic function / The logarithmic function   ($\ln x$)
- General power function(s)   ($ax^{d}$)
- Special power function(s) / Power function(s) / One-to-one power function(s)  ($x^{d}$).

Extension to quantities
The variables in the notations of the functions discussed above are meant as reals.
Exponential growth and decay can be described by functions with quantities as variables.
E.g. written as:  $x\mapsto a\,b^{x/s}$, b real >1   and   $x\mapsto a\,b^{x/s}$, 0<b<1   with x, a and s quantities, x and s of the same kind.
Names of this functions: "Exponential growth" and "Exponential decay".

The general exponential functions, as well as the functions ‘exponential growth’ and ‘exponential decay’, comply with   $f(x+s)/f(x)=f(y+s)/f(y)$  for all x, y, s (all reals, or all quantities of the same kind).  In words: this functions transform equidistant pairs into equiratio pairs. Hesselp (talk) 19:39, 13 October 2024 (UTC)

I think the lead is clear as is.—Anita5192 (talk) 20:47, 13 October 2024 (UTC)

The names are pretty standard: there is nothing to replace them with, other than making something up ourselves. I think it is proper to stress the mathematical importance and salience of THE exponential function e^x, but the (general) exponential functions are what is most used in applications. Magyar25 (talk) 12:19, 14 October 2024 (UTC)

I agree with @Anita5192 and @Magyar25 that the lead is relatively clear and that there is not need to change the terminology. Malparti (talk) 10:59, 15 October 2024 (UTC)

@Anita5192: The distinction between numer-to-number exponential functions versus the more general quantity-to-quantity ones, (the 'diverse phenomena in several sciences'), isn't made clear by the sentence "More general, especially . . . the function at that point." Hesselp (talk) 18:25, 25 October 2024 (UTC)

This looks crystal clear to me.—Anita5192 (talk) 18:41, 25 October 2024 (UTC)

@Magyar25 and Malparti: I accept that my (incomplete) proposal for naming the different types of exponential functions, isn't supported. But I still advocate a separate description in the intro of the most general type of exponential functions. Hesselp (talk) 18:25, 25 October 2024 (UTC)

These are mentioned in the lead and described later in the article, where they should be.—Anita5192 (talk) 18:41, 25 October 2024 (UTC)

Quantity-to-quantity exponential functions
The current introduction starts with the exponentiel functions of  (1) type  $e^{x}$ and  (2) type  $b^{x}$. Followed (in the sentence "More probably, especially in applied settings, ...") by a incomprehensible mix of two more types:
(3) type  $a\ b^{x}$,  mapping numbers to numbers ("$\mathbb {R} \to \mathbb {R} ^{+}$") ;
(4) type  $a\ b^{t/u}$,   mapping quantities to quantities; with argument quantity t (mostly: 'time') measured by unit quantity u of the same kind. Describing the "diverse phenomena in ... sciences."   Obeying $f(x+z)/f(x)=f(y+z)/f(y)$ for all x, y, z (transforming equidistant pairs into equirational pairs),  or  $f(x)/f'(x)$ is independent of x . With arguments and images not restricted to numbers.

Question:  Shouldn’t this quantity-to-quantity type be described explicitly in the intro, not mixed up with the description of number to number type $a\ b^{x}$ ?

Second question:  Isn't it preferable to start the intro with the most general type of exponential functions: mapping quantities to quantities?   Followed by its subspecies, with one more restriction added successively:
- type  $a\ b^{x}$, arguments and images restricted to numbers;
- type   $b^{x}$,  obeying   $f(x+y)=f(x)f(y)$ for all x, y (transforming adding into multiplication)  or obeying  $f(0)=1$  ;
- singleton type  $e^{x}$,  obeying  $f'=f$,  usually named - because of its importance - 'the exponential function'.

Then showing the rewriting of the written forms using an arbitrary (positive) base, by the often preferred forms with base e . Maybe with mentioning that the Euler number e can be defined as the x-independent value of expression  $f(x+f(x)/f'(x)\,)\ /\ f(x)$   with  f  being any exponential function, including the quantity-to-quantity type. Hesselp (talk) 18:54, 25 October 2024 (UTC)

Quantity to quantity? Mathematical objects aren't defined in terms of physical quantities, but in terms of other math objects. For example, what would we mean by "exponential of time"? Well, we measure time by a number, and compute the exponential of that number. Thus, there is no separate exponential of time, only the application of exponential of number. This is essential to the viewpoint of mathematics as a discipline.

As for the exposition progressing from special to general, versus general to special, I prefer the first, because the key function to understand is exp(x), while the others should be thought of as modifications of it. Magyar25 (talk) 20:48, 25 October 2024 (UTC)

I agree with Magyar25 that physical quantities are measured by numbers, and thus only functions from numbers to numbers are to be considered. However, when one has quantities, one must consider how formulas change when one changes of units. Here a (general) exponential function establish a relation ⁠$y=ab^{cx}$⁠ between x and y. This relation can be rewritten ⁠$y/a=e^{x/d}$⁠ with ⁠$d={\frac {1}{c\ln b}}.$⁠ This means that, when working with quantities, there is only one exponential function, since one can choose the units for having the natural exponential function.

IMO, this does not belong to the lead, but could be the object of a section "Exponential of quantities" somewhere in the article. D.Lazard (talk) 09:21, 26 October 2024 (UTC)

@Magyar25:

a.  "Quantity to quantity" exponential functions.   This aren't mathematical objects? Function (mathematics) says: "Functions were originally the idealization of how a varying quantity depends on another quantity."

b.  "we measure time by a number".   Other people measure time by a (arbitrary chosen) time interval / time unit. That' s not a number.

c.  "the key function ... is exp(x) or $e^{x}$".   But this function exp and this number e are just falling from the sky, where is their origin ?  The answer: in every function transforming equidistant pairs of domain elements into equirational pairs of codomain elements. Isn't it that the key, to start with?

d.  "the others should be thought of as modifications of it".   The function types $e^{x}$, $b^{x}$ and $a\ b^{x}$ (all numbers to numbers) are nested subclasses of the most general class of exponential functions. The decay of U235 radiation intensity as a function of time, cannot be thought as a modification of $e^{x}$.   The elements in the class of functions of type $a\ b^{x}$ are not 'modifications' of function $e^{x}$ (by the way: how do you define 'a modification of a given function' ?). You only can say that the function $e^{x}$ is an element of the class of functions of type $a\ b^{x}$.

e.  "any function $\mathbb {R} \to \mathbb {R}$ defined by . . . $a\ b^{kx}$" (intro since 23 Oct 2023 / 15 Oct 2024 ).   Why a notation with parameters b and k ?  Both being numbers, $b^{k}$ can be reduced/simplified to one parameter.

@D.Lazard:

f.  "working with quantities, there is only one exponential function".   The exponential decay function of U238 is the same as the the exponential decay function of U235 ?  I don’t think so.  Yes, they both can be written using an exponentiation form with base e and different exponents, but this partly similarity of notation does't makes them the same function/relation, IMO. Hesselp (talk) 16:18, 26 October 2024 (UTC)

The conceptual framework of mathematicians is different from that of physicists and engineers. Yes, "functions were originally the idealization of how a varying quantity depends on another quantity," but centuries of math have refined this concept to a precise abstract core defined in terms of set theory. It is only through such precision (definition, theorem, proof) that we can build the formally correct theories which are the content of modern mathematics.

Of course, there is much informal intuition behind such theories, and many ways to model real-world phenomena using them. But mathematics is not intuition or empirical science, and I believe that Wikipedia mathematics articles should guide a general audience toward the mathematics, i.e. toward the formal theories.

Regarding exponential functions, the mathematical consensus is that exp(x) is not a random function from the sky, but a function so special that it will appear inevitably in any investigation of differential equations or growth models. It is characterized in at least 5 ways, most of them leading naturally to exp(x), not to $a\,b^{x}$. Most fundamentally, $\exp '(x)=\exp(x),\ \exp(0)=1$. The decay of U238 and U235 are not mathematical functions; rather, they are modeled by $f(x)=a\exp(-kx)$ for different constants a, k. If you measure radioactive material carefully enough, you will find deviations from this model, as you will from any mathematical model. Magyar25 (talk) 17:19, 26 October 2024 (UTC)

@Magyar25: I numbered your ten sentences, for easy reference.

1. Too much a generalization. The conceptual framework of different mathematicians, can differ at least as much as between some mathematicians and some physicists/engineers.

2. In WP Function (mathematics) I cannot find that the time dependency of the intensity of U235 radiation shouldn't be called 'function'.

3. 4. Agree

5. Agree. So I expect you can define 'a modification of a given function' (I don't mean: 'modification of notations of a function'). And explain why $b^{k}$ (b, k in $\mathbb {R}$) is not reduced to one variabel.

6. My remark in point c about 'from the sky'. I meant: 'in the very first sentence of the intro', of course not the concensus between mathematicians.   I'm not at all opposing the central role of $e^{x}$ in mathematics/analysis (this central role is probably caused by the fact that $e^{x}$ obeys more conditions/restrictions than the other types of exponential functions).

7. I've no reason not to believe you.

8. Agree, see 6.

9. They are modelled as well by $f(t)=a\ b^{t/u}$ with b a positive number, t and u time intervals, a an intensity of radiation.

10. Agree. Hesselp (talk) 21:58, 26 October 2024 (UTC)

Intro reduced to essentials - proposal

An exponential function is a function obeying:  pairs of elements with the same difference in the domain, are transformed into pairs with the same ratio in the codomain  ($\,f(u{+}d)/(f(u)=f(v{+}d)/f(v)$ for all domain elements u, v, d ).  An equivalent condition is: $f(x)/f'(x)$ is independent of $x$.  The $x$-independent ratio $f(x{+}1)/f(x)$ is called its base.
Notation.  Exponential functions are usually written $a\ b^{x}$, using the notation of the two-variable function exponentiation.
Special type.  With a equals 1 ( f(0)=1), the functions $f(x)=b^{x}$ obey $f(x{+}y)=f(x)f(y)$ for all $x$, $y$  (transforming addition into multiplication, the opposit of the main property of logarithmic functions).
Most special type.  There is exactly one function obeying moreover $f(x)/f'(x)=1$ for all $x$, having Euler’s number e (2.71828…) as its base. Usually named 'the exponential function' or 'the natural exponential function'.  Symbol: $\exp$,  written as  $\exp \,$(x),  e^x or  e^x .
[End of poposed intro.]

Nine remarks:
- The order of the three types of exponential function from special to general is less easy to grisp than the order from general to special. Because: what is meant by "generalizations of a given function' (see hatnote), or with 'modification of a given function' (see Talk Magyar25, 25 Oct.)? Why aren't 'generalizations/modifications' as well:  $e^{x^{2}}$,  $e^{[x]}$,  $x\,e^{x}$,  $e^{sinx}$,  $t\mapsto a\,e^{t/\tau }$ (time t to quantities as $a$),
- All info about quantity-to-quantity relations/functions ("applied settings") in a separate section, or in Exponential growth and Exponential decay.
- Not starting with how (a special case of) an exponential function is displayed in written form, but with its defining property.
- No history in the intro ("the exponential function originated from").
- "relating exponential functions to the elementary notion of exponentation". Not the exponential function 'an sich', but the usual way they are notated is related with exponentation.
- Don't use "base" (of an exponential function) without defining the term.
- No emphasis on b^x = e^{x ln b}. Every positive number, so every value of b^x as well, can be written as an exponentiation with an arbitrary (pos.) base. So with base e as well. Belongs to article Exponentiation.
- Not: "its rate of change at each point is proportional to the value of the function at that point."  Two numbers have a ratio, but two numbers cannot be proportional.
- Postpone (or avoid):     complex arguments,     matrices,     Lie-algebras,     symbol “ln” (not simple enough for the intro),     "the natural exponential" (= the natural exponentiation? sources),     the notation a b^kx (because b^k  instead of a single variable, suggests that k stands for a quantity instead of a number),     "initial value problem",     power series definition,     square matrices,     Lie groups,     Riemannian manifold,     antilogarithm (is/was used as well for the inverse of logarithmic functions with base 10 and other bases).

Comments? Hesselp (talk) 23:13, 31 October 2024 (UTC)