Talk:Exponential function

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)

This seems confusing and overcomplicated to me. This article should focus on "the" exponential function ⁠$\exp$⁠ as its primary subject, while mentioning more general functions as as a side subject. The discussion of the latter in the lead section can be copyedited and slimmed down, but elaborated more carefully in a later section of the article. –jacobolus (t) 18:57, 1 November 2024 (UTC)

@Jacobolus: Three questions:

• "should focus on". Why 'should it'? The title doesn't support you.
• Sources please, with a definition of 'more general functions of a given function'.
• The article starts with saying how your favorite function is denoted. Where in the article I can find for the first time how your favorite can be defined? (In my proposal by sentence 1, 5 and 6, Isn't that an early 'focus'?)

Hesselp (talk) 20:42, 1 November 2024 (UTC)

Because (a) this is the current main subject of the article, (2) this is one of the most common and important functions in mathematics and an article scoped to be about it in particular is worthy of an encyclopedia article with plenty to say, (3) many people search for "exponential function" trying to figure out what it is, and most inbound links here are referring to this specific function, with text like "foo is exp of bar, where exp is the exponential function" or "⁠$x\mapsto e^{x}$⁠ is called the exponential function", (4) the more general topic is also discussed at articles such as exponential growth and exponentiation. –jacobolus (t) 02:00, 2 November 2024 (UTC)

@Jacobolus: Thanks for your reaction. But . . . I don't find answers on any of my three questions.

• Visiting linear function, power function or logarithmic function I don't see a focus on one special individual function. So why the article exponential function  s h o u l d  focus on $e^{x}$ ?
• "more general functions than function e x p". See my first 'remark' (in Talk 31 Oct).
• The current text defines function $\exp$ in sentence nr. ??. That means a stronger 'focus' than sentence nr. 6 (Most special type) in my proposal?

On your (2): start an article titled "Function $\exp$".  (3): "foo is exp of bar" ??  (4): Your "also" implies that type $a\,b^{x}$ is discussed in the current article/intro as well. So why it should be "confusing and overcomplicating" in my proposed intro?   Starting with type $a\,b^{x}$ avoids the undefined ídea of 'generalizing a function'. Hesselp (talk) 12:09, 2 November 2024 (UTC)

I'm not really understanding your point, but I strongly oppose an effort to substantially change the scope of this article. If the problem is just that the current lead section is poorly written, then I agree, and would happily support efforts to make it clearer. –jacobolus (t) 15:35, 2 November 2024 (UTC)

Yes, let's focus on exp x, whether written that way or as e^x, rather than the other forms. The lede could indicate in a single sentence that there exist other meanings, but we wouldn't actually write or discuss the likes of a^b or ca^b, etc. except in some section much later. Likewise we'd have a single sentence mentioning the existence of exp x when x is something more complicated than a complex number, but wouldn't actually write or discuss it until a later section. Excepting for these hints, I want the lede to be exclusively about exp x for x a complex number or simpler. —Quantling (talk | contribs) 20:12, 1 November 2024 (UTC)

@Quantling: "let's focus on the function written as exp x or as e^x ", followed by ". . .there exist other meanings". As if the meaning of the written forms exp x and e^x is already explained. Quod non.   Please show how you should explain the meaning of your forms in the first lines of the lede. Hesselp (talk) 21:12, 1 November 2024 (UTC)

I am thinking that we should have something like this (after the short description, about blurb, and infobox):

The exponential function is a mathematical function denoted by $f(x)=\exp(x)$ or $e^{x}$ (where the argument x is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, although it can be extended to the complex numbers or generalized to other mathematical objects like matrices or Lie algebras. Its ubiquitous occurrence in pure and applied mathematics led mathematician Walter Rudin to consider the exponential function to be "the most important function in mathematics".^[Rudin] The function can be defined in many equivalent ways, such as via a power series, an infinite product, or a differential equation. In many contexts the exponential function is the same as exponentiation, exp x = (exp 1)^x. More generally, functions of the form a^b or ca^b are sometimes called exponential functions.

Definitions

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The function exp(x) can be defined in several equivalent ways.

Taylor series

The exponential function can be defined by the power series $$\exp x=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}$$ which works in many contexts such as when x is a real number, a complex number, or a matrix. More about it...

Infinite product

The exponential function can be defined by the infinite product $$\exp x=\lim _{n\rightarrow \infty }\left(1+{\frac {x}{n}}\right)^{n}$$ which works in many contexts such as when x is a real number, a complex number, or a matrix. More about it...

Differential equation

The exponential function can be defined by the differential equation $${\frac {d}{dx}}\exp x=\exp x$$ with the initial condition that exp(0) = 1. More about it...

—Quantling (talk | contribs) 16:21, 2 November 2024 (UTC)

I think it's a mistake to lead with complex numbers, matrices, Lie algebra, or quotations from Rudin. This is a subject encountered by high school students in their calculus or even algebra classes, and the first couple paragraphs and after that the first few sections should be made as accessible as possible. See WP:TECHNICAL. –jacobolus (t) 16:51, 2 November 2024 (UTC)

Who prefers the current intro?

Having considered the comments on earlier proposals, I wrote a partly new version of the intro. Based on principles as:
- Considerable shortening
- As long is the title is not changed into "Function exp", the intro should start with a general definition condition applicable to all types of functions that are called 'exponential' by scientists. (See intro 5 Sep. 2009 – 25 Aug. 2015; see Talk 2016)
- The natural exponential function should have a prominent place.
- No mentioning of alternative definitions as power series, limiet of a sequence (it's not an infinite product !), continued fraction.
- Nothing about history, matrices, Lie algebra, Rudin, the natural exponential (=?), initial value problem, Riemannian manifold, bijection.
- Encyclopedic style, not a written classroom presentation.

Alternative intro 11-06
An exponential function, represented by written forms as $a\,b^{x}$, $b^{x}$, $e^{x}$, $\,\exp x$ and more, is a mathematical function defined by the condition that pairs with the same difference, are transformed into pairs with the same ratio.
Exponential functions can be represented by written expressions as $\,\exp x$, $e^{x}$, $b^{x}$, $a\,b^{x}$ and more.  Some authors use the name 'exponential function' for a broader class of functions than others. All this functions obey the condition that pairs of arguments with the same difference in the domain, are transformed into pairs of values with the same ratio in the codomain.  This corresponds with the example of duckweed growth: in equal long periods of time, the area of duckweed increases by the same multiple (the same factor). In math language:$\,f(u{+}d)/(f(u)=f(v{+}d)/f(v)$ for all $u$, $v$, $d$ .  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.

The unique exponential function ($\mathbb {R}$ to $\mathbb {R} ^{+}$) obeying  $f$(0) $=1$  and   $f(x)/f'(x)=1$ for all $x$,  is called  'the exponential function'  or  'the natural exponential function'. Symbol $\exp$, written as  $\exp x$ or $e^{x}$ or $e$^$x$.  Having Euler’s number e = 2.71828… as its base.

Exponential functions of type $f(x)=b^{x}$ obey $f$(0) $=1$  and   $f(x{+}y)=f(x)f(y)$ for all $x$, $y$ .  They transform addition into multiplication, the opposite of the main property of logarithmic functions.

Exponential functions of the general type  $f(x)=a\,b^{x}$ ($\mathbb {R}$ to $\mathbb {R} ^{+}$)  can illustrate two different meanings of the word 'base':  the exponential function  $f(x)=4\cdot 3^{2x}$ has base $f(x{+}1)/f(x)=\,9$,  while the expression  $3^{2x}$  has base $3$ (and exponent $2x$) .

Extending the (co-)domain to complex numbers leads to the complex exponential function $z\to e^{z}$, see § Complex plane
Extending the (co-)domain to quantities, exponential relations between quantities (in other sciences than pure mathematics) can be described. To avoid exponentiations with an invalid exponent, this functions are noted as $t\to a\,b^{t/\tau }$, with $b$ real positive (≠1), and $\tau$ the unit that measures quantity $t$ (mostly 'time'). See Exponential growth and Exponential decay.   [End of proposed intro.]

Who prefers the current intro? Hesselp (talk) 22:11, 6 November 2024 (UTC)

Although the current lead has many issues, the proposed versiom is much worse. One of the main issues is that the first sentence is much too WP:TECHNICAL and is confusing for most people who have already heard of exponential functions and want to refresh their knowledge. Also, I am not sure whether the functional equation given as a definition implies continuity and differentiability. D.Lazard (talk) 09:06, 7 November 2024 (UTC)

More: this first paragraph seems WP:original research, since I do not know any standard textbook that define exponentials this way. This is a strong reason for not accepting this proposal. D.Lazard (talk) 09:11, 7 November 2024 (UTC)

@D.Lazard: I've rewritten the first sentence of my proposal 11-06. Added an example, as advised by WP:TECHNICAL. (I hope the duckweed-in-the-pond story is well-known enough.)  And I changed 'definition' into 'condition', because you can't define THE meaning of a term when there is no consensus.   I'm not the inventor of the equal-difference-to-equal-ratio condition [Article 19 Apr. 2011 16:04, second sentence:  "a relationship in which a constant change in the independent variable gives the same proportional change (i.e. percentage increase or decrease)" ] .
You use the word 'exponential' as a noun. Shouldn't this noun be defined in the intro? Do you have a suggestion? Hesselp (talk) 18:53, 9 November 2024 (UTC)

I definitely prefer the draft proposed below by Jacobolus. Cosmetic changes of your version, such as your recent one, cannot change my opinion. D.Lazard (talk) 21:17, 9 November 2024 (UTC)

I prefer the current introduction. The exponential function is exp x. If there is an article about growing exponentially then we prominently feature ab^x there. In the article about exponentiation you can feature a^b. Almost never seen expressions like "f(u+d) / f(u) = f(v+d) / f(v) for all u, v, and d" do not belong anywhere near the introduction. Expressions like "f(x) / f′(x) is independent of x" should be written as a differential equation without any denominators. Actually, b is the base of an expression like f(x) = b^x; the value of f(x + 1) / f(x) is just one of many ways of rediscovering it. The "natural" exponential function can be interpreted as exponentiation with a base of b = e but I'd demote that to the fourth-most important way of defining it, after an infinite series, an infinite product, and a differential equation (in some order). —Quantling (talk | contribs) 13:53, 7 November 2024 (UTC)

I think the starting characterization can be something like:

In mathematics, the exponential function is the unique real function which maps ⁠$0$⁠ to ⁠$1$⁠ and has a derivative equal to its value. The exponential of a variable ⁠$x$⁠ is denoted ⁠$\exp x$⁠ or ⁠$e^{x}$⁠, with the two notations used interchangeably. It is called exponential because its argument can be seen as an exponent to which Euler's number ⁠$e\approx 2.718$⁠, the base, is raised.

The exponential function converts sums to products: The exponential of a sum is equal to the product of separate exponentials, ⁠$\exp(x+y)=\exp x\cdot \exp y$⁠. Its inverse function, the natural logarithm, ⁠$\ln$⁠ or ⁠$\log$⁠, converts products to sums: ⁠$\ln(x\cdot y)=\ln x+\ln y$⁠.

Other functions of the general form ⁠$f(x)=b^{x}$⁠, with base ⁠$b$⁠, are also commonly called exponential functions, and share the property of converting addition to multiplication, ⁠$b^{x+y}=b^{x}\cdot b^{y}$⁠. Where these two meanings might be confused, the exponential function of base ⁠$e$⁠ is occasionally called the natural exponential function, matching the name natural logarithm. The "natural" base ⁠$e=\exp 1$⁠ is the unique base satisfying the criterion that the exponential function's derivative equals its value, ⁠${\tfrac {d}{dx}}e^{x}=e^{x}$⁠, which simplifies definitions and eliminates extraneous constants when using exponential functions in calculus. In higher mathematics, the meaning of standard exponent notation ⁠$b^{x}$⁠, generalized to allow arbitrary real numbers as exponents, is usually formally defined in terms of the exponential and natural logarithm functions, ⁠$b^{x}=\exp(x\cdot \ln b)$⁠.

Quantities which change over time in proportion to their value, for example the balance of a bank account bearing compound interest, a bacterial population, the temperature of an object relative to its environment, or the amount of a radioactive substance, can be modeled using functions of the form ⁠$y(t)=ae^{ct}$⁠, also sometimes called exponential functions; these quantities undergo exponential growth if ⁠$c$⁠ is positive or exponential decay if ⁠$c$⁠ is negative.

The exponential function can be generalized to accept a complex number as its argument. This reveals a relation between the multiplication of complex numbers and rotation in the Euclidean plane, Euler's formula ⁠$\exp i\theta =\cos \theta +i\sin \theta$⁠: the exponential of an imaginary number ⁠$i\theta$⁠ is a point on the complex unit circle at angle ⁠$\theta$⁠ from the real axis. The identities of trigonometry can thus be translated into identities involving exponentials of imaginary quantities. The complex function ⁠$z\mapsto \exp z$⁠ is a conformal map from an infinite strip of the complex plane ⁠$-\pi <\operatorname {Im} z\leq \pi$⁠ (which periodically repeats in the imaginary direction) onto the whole complex plane except for ⁠$0$⁠.

The exponential function can be even further generalized to accept other types of arguments, such as matrices and elements of Lie algebras.

This is a bit rushed as I'm in a hurry. I'll try to copyedit/clean this up/extend it further later. –jacobolus (t) 21:49, 7 November 2024 (UTC)

I would prefer a definition that easily generalizes to the use of the exponential function for complex numbers, matrices, and tangent vectors. With e^x interpreted as an exponentiation with base e, we'd have to prove equivalence to a power series or infinite product representation, or that it satisfies a particular differential equation ... and then use that representation to generalize the real-valued function beyond the real numbers. Differential equations and infinite products are more advanced than power series, so I'd like to put power series before any of the others. —Quantling (talk | contribs) 22:14, 7 November 2024 (UTC)

It's not helpful in my opinion (see WP:TECHNICAL) to lead with a definition as a power series or similar, involving a big wall of mathematical notation or more advanced topics unfamiliar to some of the expected audience. We can in the 2nd or 3rd section after the lead discuss various possible definitions. My description above is a definition in terms of a differential equation, just phrased in a hopefully relatively accessible way. –jacobolus (t) 23:12, 7 November 2024 (UTC)

@D.Lazard does the first paragraph or two there seem accurate / complete enough? –jacobolus (t) 03:36, 8 November 2024 (UTC)

This proposed lead is fine for me. I like the start with the differential equation because this explains why the natural exponential is privileged among other exponential functions (no choice of other constants than 0 and 1). To editor Quantling: this definition generalizes immediately to the complex exponential by simply changing "real" into "complex" in it. The series expression is an immediate consequence of the definition of Taylor series. Also, the definition of the exponential as a series does not explain the choice of the coefficients of the series. I agree with you that differential equations can be viewed as more advanced than power series (this depends on teachers choices), but there is only derivatives, not differential equations in Jacobolus lead. D.Lazard (talk) 10:19, 8 November 2024 (UTC)

The Characterizations of the exponential function article touches on some of the topics we discuss here.

"has a derivative equal to its value" is an equation and solving such an equation is more advanced than manipulating a power series. But, as you say, it is not obvious why someone would find that power series interesting ... except for the fact that it satisfies that differential equation! So, the differential equation is good.

However, I want to stay away from defining the exponential function as b^x for a particular value of b, though I have no objection to deriving that (for some domains) the exponential function can be so characterized. —Quantling (talk | contribs) —Quantling (talk | contribs) 21:57, 8 November 2024 (UTC)

I wasn't trying to define the exponential function as a power of ⁠$e$⁠, but merely to explain where the name exponential comes from (related to "exponent"). I realized I didn't actually know too much about the history of the word "exponent", so went hunting a bit. It appears in the Lexicon Technicum several times, and seems like it started as a synonym for index; in a geometric progression the indices (or exponents) of the term can be added/subtracted in a way corresponding to the multiplication/division of the terms themselves (for example; aside, somewhat confusingly c. 1700 the term exponent also meant the numerical value of a ratio, e.g. the ratio 6:2 would have the "exponent" 3; I'm also somewhat confused about what precisely exponential curves, exponential quantities etc. were supposed to mean as found in v2). Anyway, I think it's important to point out the relation between the words "exponential" and "exponent" near the beginning of this article. –jacobolus (t) 23:32, 8 November 2024 (UTC)

I think that the history of the word exponent must be searched in Latin, where the corresponding word means "to be exposed". I guess that it was used by copists and early printers for "superscript". Indeed, in French, exposant means both "superscript" and "exponent", as well as indice means both "subscript" and "index". As the notation of exponents as superscripts was introduced by Descartes, it is also possible that "exponent" was derived from French rather than from Latin. D.Lazard (talk) 10:17, 9 November 2024 (UTC)

@D.Lazard, you may be right that just "maps ⁠$0$⁠ to ⁠$1$⁠" is clearer in the first sentence, but I do think it's worth pointing out explicitly somewhere fairly near the top of this article that 0 is the additive identity and 1 is the multiplicative identity, since this is related to the property of transforming addition in the domain to multiplication in the codomain. –jacobolus (t) 23:37, 8 November 2024 (UTC)

I suggest to modify the second paragraph from "The exponential function converts sums to products: The exponential of a sum is equal to the product of separate exponentials, ⁠..." into The exponential function converts sums to products: it maps the additive identity 0 to the multiplicative identity 1, and the exponential of a sum is equal to the product of separate exponentials, ⁠... D.Lazard (talk) 09:51, 9 November 2024 (UTC)

Sounds good. Maybe I'll insert this proposal into the article in the next day or so, if nobody has objections. The rest of the article could also use significant cleanup and expansion if anyone wants to collaborate on this. –jacobolus (t) 18:17, 9 November 2024 (UTC)

If we're emphasizing historical and if we're emphasizing the domain of the real numbers then maybe the first definition should be that the exponential function is the inverse of the natural logarithm function. That's not a chicken and egg definition because the natural logarithm for real numbers is easily defined by

$$\ln x=\int _{1}^{x}{\frac {dt}{t}}\,.$$ —Quantling (talk | contribs) 18:35, 9 November 2024 (UTC)

In my opinion this integral is worth including in section 2 or 3, but not in the first few sentences of the article. Also, I don't think "emphasizing historical" is the most important goal, though this article could certainly use a better discussion of the history somewhere. –jacobolus (t) 18:41, 9 November 2024 (UTC)

"Inverse of the natural logarithm" might be easier for the novice reader to understand than "the unique real function which maps ⁠⁠0 to 1⁠⁠ and [everywhere] has a derivative equal to its value".

Also it is apparently interchangeable in British English but, in American English, I usually see "that" instead of "which" unless it immediately follows a comma, or the sentence previously has a "that"; so it would be "... real function that maps ...". —Quantling (talk | contribs) 20:50, 9 November 2024 (UTC)

That the exponential and natural logarithm functions are inverses is definitely important to mention early, which is why I propose putting it in the second paragraph instead of at the end of the lead section. It's certainly possible to use that as a definition of the exponential function but not particularly helpful to anyone who doesn't already know what logarithms are, and any effort to explain what a logarithm is in 2 sentences is going to be just as difficult. I agree with you that having a derivative equal to its value is not immediately accessible to every reader, but I think it's better for a first sentence than a power series or other mathematical notation heavy description. Various properties / characterizations of the exponential function should be unpacked at leisure in the body of the article so readers who persist past the lead can hopefully see a bit of motivation and a few points of view. –jacobolus (t) 21:38, 9 November 2024 (UTC)

As for "which" vs. "that", this supposed rule is a bit of turn-of-the-20th-century prescriptivist nonsense which does not reflect prevailing usage in English writing or speech. See English relative clauses § That or which for non-human antecedents and https://www.merriam-webster.com/grammar/when-to-use-that-and-which. But it's entirely plausible that rephrasing this sentence would make it clearer. –jacobolus (t) 21:45, 9 November 2024 (UTC)

"rational exponential functions"

Hi @TheGameChallenger and welcome to Wikipedia. I reverted (special:diff/1256428565) the section you just added about the concept of a "rational exponential function" meaning the product of an exponential function and a rational function. In a quick literature skim I couldn't find any sources using the term "rational exponential functions" in this way, though I did find sources using that term to mean the exponential of a rational function, a rational function of an exponential, or an arbitrary function constructed from arithmetical operations and ⁠$\exp$⁠. I don't think this is a well-established term and the concept doesn't seem to be particularly widely used.

To add material to Wikipedia, you need to find "reliable sources" supporting it, and also make sure that it is on topic for the article; I think going into detail about this seems out of scope here, giving "undue weight" to a niche and tangentially related topic. –jacobolus (t) 22:34, 9 November 2024 (UTC)