Talk:Exponential function

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Graphs in 'Complex plane' paragraph[edit]

Some good graphs there, but I'm always wondering why Wikipedia (admittedly like other math sources) just ignores the comprehensive 'full 4D' graphs that do exist for most common functions, and give visual evidence for properties like periodicity, asymptotes, symmetries etc 'just in one graph'. Examples for the exponential: http://www.nucalc.com/Exponential.html and from my site: http://www.wugi.be/animgif/Expon.gif https://www.youtube.com/watch?v=hoKNb7Qk0lA&list=PL5xDSSE1qfb6FIk0Pl3VCg5p3Ema52hEG&index=5 For more graphs of the useful functions, see http://www.wugi.be/qbComplex.html https://www.youtube.com/playlist?list=PL5xDSSE1qfb6FIk0Pl3VCg5p3Ema52hEG guido 'wugi' wuyts — Preceding unsigned comment added by 81.164.174.159 (talk) 17:51, 28 October 2018 (UTC)

Decreasing exponentials[edit]

Most of the article and all the illustrations seem to assume that b >> 1, reinforcing the popular notion that "exponential" means "is growing fast" or even "has grown a lot" or even "is very big". The head should point out that an exponential function decreases if b < 1, and very quickly if b << 1 (and is constant in the borderline case of b = 1).
In particularm the graph in the head section should show at least two cases with different b > 1, at least one with b < 1, and at least one with a ≠ 1. --Jorge Stolfi (talk) 13:20, 20 January 2019 (UTC)

I have added the only (apparently) image in Wikipedia commons that contains a decreasing exponential. D.Lazard (talk) 14:34, 20 January 2019 (UTC)

... and I added remarks on coefficients (unsatisfactory), monotony, and constancy. Purgy (talk) 16:26, 20 January 2019 (UTC)

Parentheses[edit]

Sometimes there's exp(z), sometimes there's exp z. Shouldn't an article be consistent with itself? — Preceding unsigned comment added by A1E6 (talk • contribs) 17:57, 2 July 2019 (UTC)

@A1E6: For the most part, yes. MOS:MATH#Multi-letter names touches briefly on this, but is certainly not definitive. Personally, I think having parentheses for standard functions like

sin

,

exp

, etc. tends to clutter, especially when there are parens around the whole term for one reason or another. However, having looked over quite a few math articles, my general feeling is that there are a lot (although I wouldn't automatically say most, or even a majority) of editors who include them. I'm not sure if this is because it's just their preference, or because they think it's required, or maybe for other reasons. Anyway, I'd be in favor of removing them when they're not necessary. For example, it would be necessary in an expression like

\exp(x+y),

but not in

\exp x,

or even something like

\exp ix.

–Deacon Vorbis (carbon • videos) 19:41, 2 July 2019 (UTC)

Extinction coefficient?[edit]

The equations y = e^(-kx) and y = 1-e^(-kx) are extremely common in engineering and physics, but I can't find a non-specific name for k or 1/k. In electronics 1/k is a time constant like "the RC time constant". In physics k can be called attenuation coefficient and 1/k can be the attenuation length. If you change e to 2, 1/k it is called the half-life. I think I've seen 1/k called the expected or mean life, time, or distance. Can someone think of what it is supposed to be called and include it in this article? 1/k is the "expected value" but that's too general. Ywaz (talk) 12:56, 21 May 2020 (UTC)

Done I have documented this in this article and also in Exponential decay.—Anita5192 (talk) 19:47, 21 May 2020 (UTC)

Correction to my edit to the Overview section that got reverted (Forgot the binomial coefficients)[edit]

For example, using the definition $\exp x=\lim _{n\to \infty }\left(1+{\frac {x}{n}}\right)^{n}$ and $\exp y=\lim _{n\to \infty }\left(1+{\frac {y}{n}}\right)^{n}$ ,

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

49.147.83.13 (talk) 16:21, 21 May 2020 (UTC) Wondering if one could improve on that for the edit to be approved

Again, this proof in not useful without a proof of the equivalence of the definitions. If one has the equivalence, one can use the definition through derivatives: The derivative with respect to

x

of

e^{x+y}/e^{y}

shows that this function is equal to its derivative, and equals 1 for

x = 0

; thus, it equals

e^{x}

for every

x

. The proof takes only two lines and explains better why the identity is true. So, your proof is definitively not useful, except as an exercise for students. D.Lazard (talk) 16:55, 21 May 2020 (UTC)

Your proof is also incorrect (although fixable) and omits a couple things even still. I leave it as an exercise to determine where the mistake is, and also as a cautionary tale about coming up with your own proofs rather than adapting ones from existing sources. –Deacon Vorbis (carbon • videos) 17:05, 21 May 2020 (UTC)

anhducwiki

Monday, November 30, 2020