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I have been trying to find the length of an arc of an ellipse and I have been stuck with this integral for a complete day : $$\int_{0}^{x} \sqrt{a^2\cos^2t+b^2\sin^2t} dt$$ And my question is : can this integral be expressed in terms of elementary functions ? If not then does this integral have a special function or something ?

user376343
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Ahmed M. Elsonbaty
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    It is an "elliptic integral." Generally, it cannot be integrated into elementary functions except for some special values for a,b. https://en.wikipedia.org/wiki/Elliptic_integral – Doug M Jan 29 '20 at 18:06
  • @DougM forgive me for my ignorance. I'm a 15 year-old student and I have been teaching myself calculus and I don't know what is an elliptic integral so can you please tell me what would the result of this integral be ? (i mean in terms of special functions not elementary functions) – Ahmed M. Elsonbaty Jan 29 '20 at 18:09
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    @AhmedM.Elsonbaty It's a special function defined as an integral, which can't otherwise be solved in terms of elementary functions. A bit like how the function $\ln x$ can be defined as the integral $\displaystyle\int \frac{1}{x},\mathrm{d}x$, which otherwise wouldn't be expressible. – Jam Jan 29 '20 at 18:13
  • You've mentioned you were trying to compute an arc length. It's good to know that most of the time, arc length integrals can't be expressed with a closed form, even with simple functions. A list of arc lengths that are easy to solve can be found at (Question 3321398) but otherwise, you're usually out of luck and have to resort to numerical integration and special functions. – Jam Jan 29 '20 at 18:18
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    @jam The logarithm function is NOT a special function. It is considered an elementary one. – Mark Viola Jan 29 '20 at 18:23
  • @MarkViola I'm aware of that, which is why I said 'a bit like'. I was providing an analogy. – Jam Jan 29 '20 at 18:25
  • @jam Your comment seems to imply that the logarithm is only expressible as $\int_1^x \frac 1t,dt$. – Mark Viola Jan 29 '20 at 18:29
  • @jam You might be interested in THIS. – Mark Viola Jan 29 '20 at 18:33
  • @MarkViola My point was that it wouldn't be expressible in terms of simpler functions that would have been encountered by the OP thus far: polynomials, rational functions, etc. Indeed, there are other ways of expressing it but I think they would have opened up more questions. – Jam Jan 29 '20 at 18:34
  • @jam It's the inverse of the exponential function. That's elementary enough. And did you have a look at the reference I provided for you? – Mark Viola Jan 29 '20 at 20:02
  • @MarkViola I think you're missing my point that it is given the simple symbolic representation "$\ln x$" because (unlike the integral of any other integer power of $x$) it cannot be expressed with a rational function or polynomial. Not that it is nonelementary. I'm not denying that it could be expressed as something like $\exp^{-1}(x)$ but I think that's contrived and doesn't alter the fact that pedagogically it must be expressed with a new symbol, which is what I'm trying to convey. And yes, I'd read the linked page previously and am aware of nonelementary integrals. – Jam Jan 29 '20 at 20:18

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The answer to the question in the title is

No, the integral cannot be expressed in terms of elementary functions. Elementary functions are a fairly restricted group (like polynomials, the exponential function and the natural logarithm).

But you go on to ask whether it can be expressed in terms of special functions or something.

The short answer is that it is easy to get arc lengths along an ellipse in terms of an incomplete elliptic function of the second kind. But you need software like Mathematica (where it is EllipticE[]) to be able to use it freely.

The incomplete elliptic function (the complete/incomplete distinction is about whether the function deals with the complete curve or only an arc) is certainly a "special function or something".

Whether it is a special function is more debatable.

Personally, I don't think I would refer to this function as a special function. Wikipedia disagrees, so does the NIST bible. But the term "special function" is often used for a relatively small group of functions which were mainly important in physics and applied maths and intensively studied in the century or more before WW2 (usually as functions of a complex variable).

In any case the term "special function" is starting to get archaic, and I don't think it is a helpful one today. Long ago they were special in the ordinary English language sense, because they had been extensively studied and people knew how to deal with them, although they required more expertise than the "elementary" functions. [Whereas many other functions were fairly intractable except by inequalities, bounds, crude approximations etc.]

Today I would rather call something like the functions used for ellipse arc length a "named function". The reason is that it is possible to use it (and indeed many of the old special functions) with much less special expertise.

What often matters in practice today is not so much whether a function is an "elementary" or "special" function, but whether it is easy to calculate with the function using software like Mathematica.

If you want more information on how to use the complete/incomplete elliptic functions, I suggest googling. You can find plenty of tutorial material about ellipse arc lengths and the functions used to deal with them. Eg here (about halfway through he turns to elliptic integrals).

There is also a fair amount of material in Wikipedia scattered over several articles.

Added a little later

Incidentally, do not think of math software as just a way of getting numerical answers or plotting functions with known parameters (ie $\sin 2x$ rather than $\sin kx$). It is much more powerful than that. For many functions you can easily get indefinite integrals, power series etc.

I remember spending a significant chunk of my undergraduate years at Cambridge University many decades ago learning endless tricks for integrating functions. That knowledge is of little use to me now. Mathematica easily outperforms most academics on this site at integration (and certainly me, but notably not Jack d'Aurizio, whose expertise with integrals often delights me).

In general, you can cheerfully manipulate its named functions just as easily as if they were elementary functions.

I mention Mathematica just because it is widely available and good for integration. But there is a good deal of both commercial and open source software available.

almagest
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    The elliptic integrals are categorized as special functions. https://en.wikipedia.org/wiki/List_of_mathematical_functions#Special_functions –  Jan 29 '20 at 20:19
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    Yes, I noticed that. But the usage is far from universal. I don't think the term is at all helpful. – almagest Jan 29 '20 at 20:26
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    The NIST encyclopedia covers the special functions in chapters 5 to 36. https://dlmf.nist.gov/ Among them, the elliptic integrals, chapter 19. –  Jan 30 '20 at 15:24
  • @YvesDaoust (1) I know several mathematicians who work in complex variable theory. They all tend of think of special functions as things like Bessel functions or Gamma Functions, which have extensive theory related to them.. (2) the term was born in an era when the only way to get practical results for applications was to do the integrals by hand. Today what matters, and what is apparently important to the OP is whether one can (A) easily and quickly get numerical values for the functions and (B) manipulate them. In both cases practising mathematicians increasingly use software. – almagest Jan 30 '20 at 15:53
  • @YvesDaoust The downvoters and delete voters were entirely right that before my last edit, I failed to adequately answer the question as posed. That was stupid of me. I thought the "elementary" issue had been adequately dealt with in comments. But, of course, the function of comments under a Q is to get clarity on the Q. Both Qs and As should be complete in themselves. But I hope I have now corrected that. – almagest Jan 30 '20 at 15:57
  • I understand that you know better than those sources. –  Jan 30 '20 at 16:11
  • Come off it! That is not what I am saying. You know well that there is no standards body for maths. You cannot settle math issues by appeal to authority. When it comes to terminology, it is notorious that in many parts of math there are variations. Sometimes these persist for years. Sometimes fairly standard terminology changes over time. In this particular case, I do not like the implications of the word "special". It has all the wrong connotations and easily misleads. But I will edit to include your point about NIST. I will also tone down words like "more often". – almagest Jan 30 '20 at 16:23
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    @YvesDaoust It is unfortunate that we are both getting heated about an essentially trivial matter (the precise meaning of "special function"). You are one of the people on this site whose contributions I greatly respect. If you would like me to delete this answer as a favour to you, I will do so. In any case, I apologise for the first three words of my last comment. – almagest Jan 31 '20 at 03:21
  • Don't worry, we have the right to have different opinions and defend them. Excuse my previous comment. –  Jan 31 '20 at 08:30
  • @YvesDaoust Thank you. One thing this has clearly demonstrated to me as a relative newcomer is that the process can work. Whatever this answer's current shortcomings, it seems indisputable that it started out much worse and you successfully got me to make much needed improvements. – almagest Jan 31 '20 at 16:48
  • I didn't plan to comment again, but your last message made me check with Wolfram "by accident". https://www.wolframalpha.com/examples/mathematics/mathematical-functions/special-functions/. My purpose is certainly not to be right by all means, but now I am really curious to know why you consider that the expression "special function" is narrow and archaic. –  Jan 31 '20 at 17:30