Proof that sinc function is not elementary?

[pierce15]

Hey, does anyone know of a proof that the sinc function

$$Si(x) = \int \frac{\sin x}{x} \, dx$$

is not elementary? Or is it not proven?

Thanks

 

[pwsnafu]

See Risch algorithm. I don't know if there is a simpler way, but it wouldn't be unproved.

Edit: I'll just point out that "sinc", i.e. the cardinal sine, is $\frac{\sin x}{x}$ is elementary. Its antiderivative is called the sine integral.

 

[pierce15]

Is it just me or is the wikipedia page lacking? Under the examples, it doesn't show how Risch's algorithm is used, or what it even is.

 

[Citan Uzuki]

A full proof can be found in this paper.

 

[jackmell]

A full proof can be found in this paper.

It's hard. Requires a plow. And I think he should have said, "as elementary as the subject matter allows" which is not too elementary but that is life. Still though, would be nice if someone could make it simpler and easier to understand for Calculus students because the subject comes up often. Say a description that could be included in a first-year Calculus book that is likely to be understood intuitively by the student, like one whole chapter devoted to the matter.