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I am looking at my double integration example problems and I see a note that says integral of $e^{-x^2}$ is not an elementary function. What does that mean? And why isn't that an elementary function?

It actually says observe integral of $e^{-x^2} dx$ is not an elementary function. But why is that? Same goes for integral of (sin(x)/x) dx, why isn't that an elementary function also?

Joshua Ree
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  • Probably it should be the error function $\ \int e^{-x^2},\ $ not $\ ye^{-x^2}.\ $ The error function is often mentioned as an example of a nonelementary integral since the proof is relatively simple. – Math Gems Apr 15 '13 at 03:57
  • Ya, it actually says observe integral of $e^{-x^2} dx$ is not an elementary function. But why is that? Same goes for integral of (sin(x)/x) dx, why isn't that an elementary function also? – Joshua Ree Apr 15 '13 at 04:13

2 Answers2

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The elementary functions form the smallest differential field that contains the complex constants and the variable $x$ and such that:

  1. If $f' = g'/g$ where $g$ is elementary then $f$ is elementary (i.e. $\log(g)$ is elementary)
  2. If $f' = g' f$ where $g$ is elementary then $f$ is elementary (i.e. $\exp(g)$ is elementary)
  3. If $P(f) = 0$ where $P$ is a nonconstant polynomial with elementary coefficients then $f$ is elementary.

The theory of integration in elementary functions dates back to Liouville in 1833.

For more, see e.g. this sci.math article

Robert Israel
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  • -1 He is a student in multivariable calculus, your assuming way too much when you start talking about differential fields. This answer likely won't help him. – Eric Apr 15 '13 at 05:22
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    This answer was useful to me, though; I'd never known how elementary functions are given a precise definition. – littleO Oct 01 '13 at 05:21
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An elementary function is a function built from any combination of $+,-, \cdot, /$ and function composition with polynomial functions, exponential functions, logarithms, and the trigonometric functions of one variable.

It is very likely they meant to say the error function

$$ \mathrm{erf}(x) = \frac{2}{\pi} \int_0^x e^{-t^2}\ dx. $$

The proof that this function is not elementary (the proof would essentially show that one cannot represent the integral as a finite amount of elementary functions) is quite involved. The author of the book is likely trying to demonstrate the integration of an elementary function ($e^{-t^2}$ is elementary) does not necessarily yield and elementary function as the result. One more time,

$e^{-t^2}$ is elementary but $\int e^{-t^2}$ is not elementary.

Community
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Eric
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