Talk:Exponential function/Archive 2

Archive 1

Archive 2

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

---

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)

"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)

@Jacobolus ok i see thx TheGameChallenger (talk) 22:36, 9 November 2024 (UTC)

@TheGameChallenger Do you have sources related to this, or is it just something you worked on yourself? –jacobolus (t) 23:20, 9 November 2024 (UTC)

@Jacobolus I was working on a problem having to do with repdigits, and found that it was close to an exponential function, but not quite. I then realized it could be expressed as an exponential and rational function combined. I had known that empirical data will always be off from an exact function, but was suprised when it occurred with pure math. I then thought that if it occurred with pure math, then it may be useful for finding even more accurate analytical representations of real-world phenomena and thus wrote the page. But I now understand that its very niche and has limited uses and sourcing, so thanks for letting me know. TheGameChallenger (talk) 00:15, 10 November 2024 (UTC)

I'm sure similar functions appear in many applications. I just don't think this deserves extensive coverage in this article, among other reasons because (a) I don't think it reveals too much about the exponential function per se and (b) there are many ways we could make up specific combinations of exponential + other kinds of functions in one way or another, and there's definitely not space to cover those all in depth here, so trying to do it neutrally would balloon the article scope. But if you made some interesting observations it's worth trying to publish them at some other venue. If you can find several reliable sources discussing a particular other type of function not currently discussed in Wikipedia and giving it a common name, it would even be possible to make a new Wikipedia article about that. (Also, I hope this doesn't discourage you from making other contributions to Wikipedia – not trying to scare you off.) –jacobolus (t) 03:06, 10 November 2024 (UTC)

okay thx, I see your point now. But adding stuff to Wikipedia seems really hard, with all the sourcing and everything. The most I've added at once is just a paragraph in a larger article about video game preservation. TheGameChallenger (talk) 03:49, 18 November 2024 (UTC)

Draft of a section "Fundamental properties"

In a preceding thread, Jacobolus proposed a new version of the lead in which the exponential function is first defined as a function equal to its derivative. Some editors challenged this choice, arguing that it uses a more advanced knowledge than other definitions. IMO, the best definition is a definition that non only requires the less prerequesties, but also allows the easiest derivation of the fundamental properties of the subject. Jacobulus' first satisfies clearly this criterion, but, when trying to show this, I remarked that there is presently no section "Properties" in the article. So, I wrote a draft (below) for such a section, which shows how easy it is to deduce every properties of the exponential function from the above definition (all proofs are given in the draft, and need each less than a line).

Adding the draft to the article cannot be done immediately, because (1) it must be discussed and improved first (2) Jacobolus' lead must be implemented first (3) adding this section requires to modify a large part of the remainder of the article.

Here is the draft:

The exponential function is the unique differentiable function that equal its derivative, and takes the value 1 for the value 0 of its variable.

This definition requires a uniqueness proof and an existence proof, but it allow an easy derivation of the main properties of the exponential function.

Uniqueness: If ⁠$f(x)$⁠ and ⁠$g(x)$⁠ are two functions satisfying the above definition, then the derivative of ⁠$f/g$⁠ is zero everywhere by the quotient rule. It follows that ⁠$g(x)$⁠ is constant, and this constant is 1 since ⁠$f(0)=f(0)$⁠.

Existence as the inverse function of the natural logarithm: The inverse function theorem inplies the natural logarithm has an inverse function that satisfies the above definition.

Series expansion: The above definition implies immediately that the MacLaurin series of the exponential function is $${\begin{aligned}\exp(x)&=1+x+{\frac {x^{2}}{2!}}+\cdots \\&=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}},\end{aligned}}$$ where $n!$ is the factorial of n (the product of the n first positive integers). Taylor theorem implies that this this series is convergent for every x. The fact that the exponential function is the sum of its Taylor series results from the fact that the series equals its formal derivative, and thus its sum satisfies the above definition.

Functional equation: The exponential functions satisfies the identity $$\exp(x+y)=\exp(x)\cdot \exp(y).$$ This results from the uniqueness and the fact that the function $f(x)=\exp(x+y)/\exp(y)$ satisfies the the above definition.

Positiveness: The exponential function is positive and monotonically increasing. The latter property results from the first one, since the derivative equals the function. The positiveness results for $x>0$ from the fact that all terms of the above series are positive. For $x<0,$ this results from the functional identity that implies $\exp(-x)=1/\exp(x).$

Extension of exponentiation to positive real bases: Let b be a positive real number. The exponential function being the inverse each of the other, one has $b=\exp(\ln b).$ If n is an integer, the functional equation of the logarithm implies $$b^{n}=\exp(\ln b^{n})=\exp(n\ln b).$$ Since the right-most expression is defined if n is any real number, this allows defining ⁠$b^{x}$⁠ for every positive real number b and every real number x: $$b^{x}=\exp(x\ln b).$$ In particular, if is the Euler's number $e=\exp(1),$ one has $\ln e=1$ (inverse function) and thus $$e^{x}=\exp(x).$$ This shows the equivalence of the two notations for the exponential function.

This will be completed later, but it seems sufficient to show that this approach seems the simplest for a comprehensible presentation of the main properties of the exponential. D.Lazard (talk) 12:09, 10 November 2024 (UTC)

In case you see a way to extend this without overcomplicating it ... that ⁠$\exp(n\ln b)$⁠ can be extended from integers n to non-integers n is more general than to real numbers n. That is, the expression ⁠$\exp(n\ln b)$⁠ works even when n is complex or even a matrix. (This is in contrast to generalizing the domain of b which is more complicated, can be multi-valued, ....) —Quantling (talk | contribs) 17:49, 11 November 2024 (UTC)

Consider writing ⁠$\exp 1$⁠, ⁠$\exp x$⁠, and ⁠$\exp y$⁠ instead of the corresponding versions that use parentheses. —Quantling (talk | contribs) 17:52, 11 November 2024 (UTC)

@D.Lazard: Please explain (or correct) after Uniqueness: "It follows that $g(x)$ is constant."  and  "since $f(0)=f(0)$".
And after Functional equation the plural "functionS" is a typo? Hesselp (talk) 16:16, 12 November 2024 (UTC)

Partially completed. D.Lazard (talk) 15:35, 10 November 2024 (UTC)

It's also worth adding the continuous limit of compounding interest to this, i.e. $$\exp x=\lim _{n\to \infty }\left(1+{\frac {x}{n}}\right)^{n}.$$

I wonder whether it would be worth calling this something like "Characterizations and fundamental properties", and then elaborating a bit further on each of these, or whether it's better to make a concise version toward the top and then add later sections unpacking them further. –jacobolus (t) 16:57, 11 November 2024 (UTC)

"It's also worth adding the continuous limit of compounding interest to this": it is my intention to add is. However the simplest proof that I know is less simple than for other properties (it consists of applying Taylor's theorem to the logarithm of the formula). D.Lazard (talk) 17:13, 11 November 2024 (UTC)

Yes I agree; regardless of which we end up using first, in short order all of the inverse of the natural logarithm, MacLaurin series, continuous interest, differential equation, and ⁠$(\exp 1)^{x}$⁠ should be featured prominently. As for equivalence ... the polynomial ⁠$\left(1+{\frac {x}{n}}\right)^{n}$⁠ can be written out exactly using binomial coefficients. We can then show that the coefficient of any ⁠$x^{k}$⁠, which is ⁠${\binom {n}{k}}\left({\frac {x}{n}}\right)^{k}$⁠, tends to the MacLaurin series coefficient for ⁠$x^{k}$⁠ as ⁠$n\rightarrow \infty$⁠. (We could probably skip the part about how absolute convergence or some similar criterion justifies this approach to showing equivalence.) —Quantling (talk | contribs) 17:35, 11 November 2024 (UTC)

I agree that ⁠⁠$\exp(x)=\exp(1)^{x}$⁠ is fundamental, but it is more a definition of exponentiation with real exponents than a property of exponentiation (except for integer exponents, or, after some work, for rational exponents). For the limit of ⁠$(1+x/n)^{n}$⁠, your suggested proof is correct, but it needs some competence in combinatorics and a computation that is not specially illuminating; the use of Taylor's theorem on the logarithm seems thus better.

Also, I intend to add a paragraph on exponential growth and exponential decay. This paragraph must contain the property that the exponential is greater and increases faster than every polynomial, for sufficiently large ⁠$x$⁠. For the moment I have not yet found a sufficiently simple proof (searching that is not WP:OR since, if a simple proof exists, it has certainly been published a long time ago). D.Lazard (talk) 18:20, 11 November 2024 (UTC)

It maybe wins on "short" though perhaps not "simple": for a student who knows enough calculus ... the ratio of an exponential to a polynomial of degree k can be analyzed with k applications of L'Hopital's rule to show that the exponential grows faster. —Quantling (talk | contribs) 18:25, 11 November 2024 (UTC)

In a section near the top it's probably not necessary to prove the relations between these in detail inline in the text – they could be e.g. further down the page, relegated to a footnote, or even left in Characterizations of the exponential function (though that article is kind of a mess). I also wonder whether we can write a section like this using narrative paragraphs, instead of a somewhat choppy quasi-list. (It's always a bit of a trade off between orderly structure vs. smooth flow, and author/reader preferences vary, but I think for the broadest audience we benefit by aiming for breezy.) –jacobolus (t) 17:56, 11 November 2024 (UTC)

I have installed Jacobolus' lead and an improved version of the above draft. IMO this improves the article by providing a synthetic presentation of the main things that users must know for using the exponential function.

Much work remains for updating the remainder of the article and adding a section on exponential growth and decay. D.Lazard (talk) 20:47, 22 November 2024 (UTC)

I've started a separate draft at user:jacobolus/exp, but don't consider its pieces ready for inserting into the article yet, and it has a long way to go. –jacobolus (t) 00:44, 23 November 2024 (UTC)