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How does one distinguish between the natural base, usually indicated with "e" and a coefficient that, due to historical reasons, is referred to as "e" (such as in Weibull functions for dose-response)?

Bryan
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    You can't. If for some reason they have to appear in the same equation, you must rename one of them. – TonyK Jul 21 '24 at 15:16
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    +1 for a question that unfortunately needs answering sometimes. To take a physicist's example, $\mathrm{e}^{eV/(k_BT)}$, with $e$ the charge of a proton. – J.G. Jul 21 '24 at 15:32
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    The sum of a bunch of phase shifted currents might be $i = \sum_i, i_ie^{i\phi_i}$, which seems clear enough. – A rural reader Jul 21 '24 at 15:35
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    @Aruralreader Which is part of the reason people often replace $i$ as in $i^2=-1$ with $\mathrm{i}$ or $j$. If they're quirky, they might even use $\iota$. – J.G. Jul 21 '24 at 15:58
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    Context! There exist different words that are spelled exactly the same. How to tell them apart? – Somos Jul 21 '24 at 18:44
  • One approach is to use bold/italic style variants: e, e, e, and *e*. But it's probably less confusing to rename one of the e's. – Dan Jul 22 '24 at 16:39
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    @Aruralreader: Electrical engineers tend to use $j$ for the imaginary unit to avoid conflicting with $i$ for current. – Dan Jul 22 '24 at 16:43

2 Answers2

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I searched the internet for "Weibull" functions for dose-response and found Christian Ritz's GitHub reference for Weibull model functions, which demonstrates a technique that seems reasonable: Since $e$ is the standard name of a parameter, it can't be used for the base of the natural exponential function. As such, $\exp$ is used instead of $e^\cdot$ to represent the exponential function, as in:

The four-parameter Weibull type 1 model ('weibull1') is $$f(x)=c+(d-c)\exp(-\exp(b(\log(x)-\log(e))))\text{.}$$

Mark S.
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In printed mathematics, the natural base should be set in roman type: $\mathrm e$, while symbols representing quantities that can vary (depending on context or units) should use the usual italic font: $e$.

(Added in response to comment) In general, roman type should be used for all symbols in mathematics that have a fixed meaning: $\mathrm{i,j,k}$ for the base elements of the complex numbers and quaternions; $\exp,\ln,\cos,$ and so on for standard functions; $\lim, \sup,\operatorname{argmin}, ...$ for standard operations; the differential symbol $\mathrm d$; the base symbols $\mathrm{J}$ for the Bessel function; $\mathrm C^n(X)$ for the real functions on $X$ with continuous $n$th derivative; $\mathrm{L,H,W}$ as the base symbols for the function spaces of Lebesgue, Hardy, Sobolev, etc. Notice also that the number realms $\Bbb{N,Z,Q,R,C,H}$ are always set in upright, not italic, type.

John Bentin
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    Thank you so much. I always feel like I am the only one who prefers $\int_a^b\mathrm e^{it},\mathrm dt$ over $\int_a^be^{it}dt$. – Hagen von Eitzen Jul 21 '24 at 15:24
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    But I would not recommend this as a way to distinguish the two meanings within a single formula. That is asking too much of the reader's eyesight! – TonyK Jul 21 '24 at 15:28
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    Or write $\mathrm{e}^x$ as $\exp x$. – J.G. Jul 21 '24 at 15:31
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    I only downvoted because I agree with @TonyK—we shouldn't trust this mere font difference to distinguish between the two mathematical objects, so this isn't the best answer to the OP's specific question. (Personally I prefer the usual italic e to the roman e in math formulas, but that alone isn't a reason to downvote :) ) – Greg Martin Jul 21 '24 at 16:26
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    From the moment somebody writes this down, the difference disappears, hence my downvote. – Dominique Jul 22 '24 at 06:56
  • Personally I don't like it for all the reasons given, but in the UK there is or was some official standard which prescribed this usage. So I think it's wrong to downvote the answer. – ancient mathematician Jul 22 '24 at 07:43
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    Would you use an upright lowercase roman for the constant relating the diameter of a circle with its perimeter, or is "$\pi$" ok? – Didier Jul 22 '24 at 10:22
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    @Didier : The default greek font in TeX is italic for lowercase and roman for uppercase. It is possible to get italic greek uppercase in TeX (which should be used, where appropriate, for variables)—but, AFAIK, not roman greek lowercase. The workaround is to type π followed by a semicolon, not between dollar signs. I can't actually type the aforementioned semicolon here in the comment composition box, because the consequent six-character string just yields a blank. However, in the box for questions or answers, it yields a perfect roman lowercase pi. – John Bentin Jul 22 '24 at 13:54
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    @Didier : For a roman pi, the typed string π is shorter than π and gives the same result. – John Bentin Jul 22 '24 at 14:10
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    @JohnBentin My question was not "How do you do this?" but a genuine "Would you?" – Didier Jul 22 '24 at 15:03
  • @Didier : I probably would (because I'm a bit obsessional about such things) if pi occurred only a few times in a simple context. Otherwise, as is the usual case, I would give up and type the same (italic) $\pi$ as everyone else. I don't even know whether \text{π} would work between dollar signs. I should do some experiments. Incidentally, the remark about yielding a blank in my previous comment was true at the time but seems to be no longer valid. – John Bentin Jul 22 '24 at 15:28
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    @JohnBentin Thanks! I would personally not, because I'm lazy and not too rigid about this (even though I always use straight d for differentials in my papers). Apparently, the upgreek package exists, see this \ Presumably someone saw your message and fixed the issue. Seems like extended discussions are sometimes useful! – Didier Jul 22 '24 at 15:38
  • i, j, k are also used for the component representation of vectors in 2-D and 3-D Euclidean space. Related to quaternions but not the same thing. – Spencer Jul 22 '24 at 16:28
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    @Spencer : More usually, boldface $\bf{i,j,k}$ . – John Bentin Jul 22 '24 at 17:22
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    @JohnBentin: I would be very surprised if TeX supported π. Maybe MathJax allows it, but I don't think you can do that in a "real" publication. You would probably have to use XeTeX (and then argue with the publisher who is using some hilariously outdated TeX stack that doesn't speak Unicode). – Kevin Jul 22 '24 at 18:30
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    The claim “the natural base should be set in roman type” is simply wrong: setting it in italics is by far the more standard convention, both in current and pre-LaTeX typesetting, and borne out by for instance the AMS style guide. A few authors choose roman, and there are reasonable arguments for that — but this answer makes it sound like the standard normative convention to follow, and that’s just not true. – Peter LeFanu Lumsdaine Jul 23 '24 at 10:23