Talk:Exp x equals e^x

Sorry, but I don't see the point to this page. Euler's number is defined as that number which is such that $e^x = \exp x$ in the general case, and the proof that the (1+1/n)^n expression gives e is done on another page. So demonstrating that elementary algebraic manipulation on just integers seems a bit limiting.

However, the expression:

$\displaystyle \lim_{n \to +\infty}\left({1 + \frac x n}\right)^n = \lim_{\left({n/x}\right) \to +\infty}\left (\left ({1 + \frac 1 {\left({n/x}\right)}} \right)^{\left({n/x}\right)}\right)^x$

looks quite neat, and might be usable in the page Definition:Exponential Function/Real/Limit of Sequence. But it would need to be made rigorous and demonstrated to apply to the domain of real numbers, not just $\N$. --prime mover 14:26, 12 January 2012 (EST)

Well I was thinking more of the first definition given here: Definition:Euler's Number, not the third or fourth. That's what I thought would be the point of the page, just another perspective on the consistency of definitions.

Yes I know you were. As such (when the page is complete), link to it from Definition:Exponential Function/Real/Limit of Sequence. But note it needs to be proved for all reals.

I can see what you're trying to accomplish, but unless the natural number proof forms the basis of the real proof, it doesn't prove very much at all. --prime mover 17:23, 12 January 2012 (EST)

As for non-$\N$ indices, it can be done, but I would have to leave it as a {{stub}}. I think it's possible to do what you're asking me to do because I had a correspondence with dr math, I'll quote a part of it. --GFauxPas 14:48, 12 January 2012 (EST)

...so that $a^{1/2}$ should equal $\sqrt a$. Similarly, you can define $a^{m/n}$ as $n$th root of $a^m$ or ($n$th root of $a$)^$m$, but defining $a^\pi$ is still a challenge. You could define it through a limiting process: if $r_n$ is a sequence of rational numbers which converge to $\pi$, then you would hope that $\lim_{n \to \infty} a^{r_n} = a^{\pi}$, but there's a lot of work involved in proving this, showing that the result is independent of the particular sequence of rationals used, etc. Even after you have shown that this definition makes sense, you still have to show that the function $f(x)=a^x$ is continuous and then differentiable. Showing differentiability involves showing that the limit $\lim_{h \to 0} \frac {a^h - 1} h$ exists and determining its value. So there is a lot of work involved in verifying all the properties you want in exponential functions if you start from scratch. - Dr Fenton

Delete?

Don't necessarily have to delete it - but it definitely need to get it to work for real numbers. The existing natural-number approach would be worth keeping if it provides a basis for the real number proof. --prime mover 17:53, 12 January 2012 (EST)

Using this approach, continuity of $x\mapsto a^x$ is necessary. If it holds (might be circular) then the proof works immediately for reals as well. Otherwise, a completely different approach is likely required. --Lord_Farin 18:12, 12 January 2012 (EST)

What about keeping it circular for now and putting it under an "intuition" section of the limit definition? Then, if someone figures out exactly how to make it rigorous for all reals without being circular, we can give it it's own page. Similar to "Thus $\exp x$ can be (and frequently is) written and defined as $e^x$. Unless I'm misunderstanding the "can be defined as" part. --GFauxPas 09:40, 13 January 2012 (EST)