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What is the difference between closed-form expression and analytic expression?

I frequently observe them being mentioned in situations where, in my view, they are essentially equivalent.

What are some concrete distinctions between them, such as something one can do but the other cannot?

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  • $\begingroup$ The reference-request tag seems appropriate here (unless you just want impressions from different people). $\endgroup$ Commented Apr 19, 2016 at 21:52

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We wouldn't say "closed-form" for the solution of a differential equation, neither "analytic expression" for the solution of a difference equation (at least this is my impression). On the other hand, most probably, in this particular example we would use them interchanged!

But yes, I would say that they are essentially interchangeable. I would use also "explicit form" (again essentially interchangeable with those that you mention).

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  • $\begingroup$ Since there is literally no one else answering, could you spice up what you mean by this? Where and in what context do you see the 2 terms used. And could you like give an example of why it wouldnt matter? Im a math freshman, and essentially I am super unsure of my own opinions and footing on matters like this $\endgroup$ Commented Apr 21, 2016 at 0:21
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    $\begingroup$ I would say that any answer to your question would largely be a matter of opinion (mine included!). It also depends on the areas, on each particular topic, and most probably on each particular person. My advise to you is that you concentrate more on the content than on the form, and that you simply try to express the content of anything that you write in the best possible form (in your own view!!!). But simplifying: it should be a reasonable rule to use "closed form" for something of discrete type, and "analytical form" for something of continuous type (but really interpreted loosely). $\endgroup$ Commented Apr 21, 2016 at 0:41
  • $\begingroup$ It's also not completely clear which things count as closed form and which things don't. An interesting discussion of the topic is available in Robert J. Marks, "Solution of the Grazing Goat Problem: A Conflict Between Beauty and Pragmatism". journals.blythinstitute.org/ojs/index.php/cbi/article/view/91 $\endgroup$ Commented Oct 8, 2021 at 13:14
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Wikipedia has a table comparing these two terms and more. The most salient point is that the analytic moniker allows a convergent infinite series.

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    $\begingroup$ Don't understand why you got downvoted, I'm not an expert, but I definitely have worked with convergent continued fraction solutions which are analytic but not closed-form (Goodyear's or Shepperd's solution to Keplerian orbital propagation and computation of the Jacobian). And which agrees with that Wikipedia table. $\endgroup$ Commented Jan 20, 2023 at 17:07
  • $\begingroup$ @lamont, thanks for the confirmation from personal experience! We cannot understand everything or everybody in this world, but we can make a sincere effort to make it slightly better. $\endgroup$ Commented Jan 3, 2024 at 22:36
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I also think they are synonyms.

A solution to a problem that can be written in "closed form" in terms of known functions, constants, etc., is often called an analytic solution.

https://mathworld.wolfram.com/Analytic.html

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