can this integral be expressed in elementary functions?

Question

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 ?

Answer 1

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.

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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.

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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.