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It seems as though, in my analysis and calculus courses, in particular, a common cop-out when asked to prove an identity involving $e$, is the phrase "it's true by definition".

So, I'm trying to find as many definitions of $e$ in order to see just how many of these identities can actually be a definition of $e$.

So far, I've got the following (which are the ones most mathematicians know):

  • $e:=\lim\limits_{n \to \infty}(1+\frac{1}{n})^n=\lim\limits_{h \to 0}(1+h)^{1/h}$
  • $e:=\sum\limits_{n=0}^{\infty}\frac{1}{n!}$
  • $e$ is the global maximum of the function $x^{1/x}$
  • $e$ is the real number satisfying $\int\limits_{1}^{e}\frac{1}{x}dx=1 \iff \begin{cases} \frac{d}{dx}[e^x]=e^x \\ \\ e^0=1 \end{cases} $

Does anyone have any more to add to the list?

Thanks!

beep-boop
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  • @barakmanos Already got it! – beep-boop Jun 14 '14 at 14:36
  • I think most mathematicians consider the first one to be a theorem; it's only presented as a definition to beginners. Likewise, the third one is certainly not a definition. If you're looking for a list of "all identities involving $e$, then this is an impossibly broad and sort of useless question. If you're looking for just the accepted definitions, I think two and four are it. This amounts to "how many ways are there to define the exponential function", since this is the theoretical perspective on where $e$ comes from. – Ryan Reich Jun 14 '14 at 14:37
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    $e = \lim_{n\to\infty} \frac{n}{\sqrt[n]{n!}}$ – barak manos Jun 14 '14 at 14:38
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    In fact, here is all of them: http://en.wikipedia.org/wiki/Representations_of_e – barak manos Jun 14 '14 at 14:39
  • @barak manos: I would not assume that Wikipedia gives every possible definition of $e.$ I'm sure that given any finite list of definitions of $e,$ someone could cook up a new one that was not on the list. – Geoff Robinson Jun 14 '14 at 15:10
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    @Ryan Reich: I am one of the mathematicians who thinks that the first definition in the list ( at least the $\lim_{n \to \infty}$ version) is a perfectly good way to define $e$. I agree about the real number version: I'm not sure how you define $(1+h)^{1/h}$ if $1/h \not \in \mathbb{N}$ if you don't already have some sort of $\log$ and exponential functions available. – Geoff Robinson Jun 14 '14 at 15:17
  • See also characterizations of the exponential function, containing proofs of the equivalences of its various definitions. – Lucian Jun 14 '14 at 16:25

3 Answers3

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You really should be looking for definitions of the exponential function $e^x$, not definitions of $e$. Here are the most important ones that come to mind:

  1. $e^x$ is the unique function $f(x)$ satisfying $f'(x) = f(x)$ and $f(0) = 1$.
  2. $e^x$ is the inverse of the function $\displaystyle \ln x = \int_1^x \frac{dt}{t}$.
  3. $e^x$ is the power series $\displaystyle \sum_{n \ge 0} \frac{x^n}{n!}$.
  4. $e^x$ is the limit $\displaystyle \lim_{n \to \infty} \left( 1 + \frac{x}{n} \right)^n$.

I say this because once you start thinking about $e^x$, which is by far the more fundamental object, and not $e$, the relationship between the definitions becomes much more transparent. Here are short sketches of proofs that the definitions above are all equivalent:

$1 \Leftrightarrow 2$: if $\frac{d}{dx} g(x) = \frac{1}{x}$ then

$$\frac{d}{dx} g^{-1}(x) = \frac{1}{g'(g^{-1}(x))} = g^{-1}(x).$$

Conversely, if $\frac{d}{dx} g(x) = g(x)$ then

$$\frac{d}{dx} g^{-1}(x) = \frac{1}{g'(g^{-1}(x))} = \frac{1}{g(g^{-1}(x))} = \frac{1}{x}.$$

$1 \Leftrightarrow 3$: if $f'(x) = f(x)$ then $f^{(n)}(x) = f(x)$ for all $n$, hence $f^{(n)}(0) = f(0) = 1$, so the Taylor series of $f(x)$ has all coefficients equal to $1$. Conversely, the function with that Taylor series is its own derivative and satisfies $f'(0) = 1$ using the fact that power series are term-by-term differentiable inside their interval of convergence.

$1 \Leftrightarrow 4$: $\displaystyle \lim_{n \to \infty} \left( 1 + \frac{x}{n} \right)^n$ is the result of using Euler's method to compute $f(x)$, where $f$ satisfies $f'(t) = f(t)$ and $f(0) = 1$, as the step size $n$ goes to $\infty$.

(My personal opinion is that the first definition is the most fundamental one; in general uniqueness statements are very powerful. For example, there is a very short proof using the first definition, which I invite you to find, that $e^{x + y} = e^x e^y$.)

Qiaochu Yuan
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  • $e$ is the unique real number that makes the function $e^x$ (defined by extending $e^{a/b}$ by continuity) satisfy property (1).
    • – Jack M Jun 15 '14 at 07:28
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    Also, as an extra 2 cents, once you have the exponential function, I find a more natural definition of $e$ is as the constant ratio $f(x+1)/f(x)$, rather than as $f(1)$, which makes it seem a little arbitrary ($f(1)$ exists for all functions $f$, but the fact that that ratio is constant is a remarkable property of the exponential function). – Jack M Jun 15 '14 at 07:31
  • I would say that we shouldn't define the exponential function by the differential equation, because you would need to prove that it exists, which is most easily done by the definition via the power series. I would however agree that the differential equation is the one that lies at the heart of the exponential function. Also, the power series works for complex numbers, while the second definition would require a line integral in the complex plane and Cauchy-Goursat theorem or something. – user21820 Dec 04 '14 at 05:22
  • @user21820: I think it's important to distinguish between definitions and constructions. I agree that it's very convenient to construct $e^x$ via power series, but the fundamental reason why you're looking at that power series and not some other power series is because it solves a differential equation that you care about. I am fine with definitions including theorems that the definitions are well-defined. – Qiaochu Yuan Dec 04 '14 at 05:46
  • @QiaochuYuan: Yup I agree, but this point is not clear to many students, because few people point out to them precisely what kind of definitions are allowed, and so they easily fail to see when something is not well-defined. For that reason, I normally introduce the exponential function by seeing what a solution to the differential equation must look like if it existed and can be expressed as a power series around 0, and then I prove that the power series actually converges everywhere and does satisfy the differential equation. – user21820 Dec 04 '14 at 05:57
  • @QiaochuYuan How would you prove $e^{x+y} = e^x e^y$ based on the first definition $\left( \left(e^x \right)' = e^x \right)$ pls? – athos Sep 29 '22 at 17:51
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    @athos: $e^{x+y}$ and $e^x e^y$ are both solutions to the ODE $f'(x) = f(x)$ with initial condition $f(0) = e^y$. So by the uniqueness theorem they must be equal. – Qiaochu Yuan Sep 29 '22 at 17:51