Why can't $e^{x^2}$ be integrated

Question

My teacher told me that not only do we have to use the erf function to approximate error, but that it is proved impossible to integrate in real analysis (at least not Riemann-integrable). Is there a name for this proof, and can I have it? I am not a mathematician, but such a simple function has been around and it irks me terribly and need a detailed proof for closure.

Have a look at http://math.stackexchange.com/questions/523824/what-is-the-antiderivative-of-e-x2 — pitchounet, Jan 23 '14 at 09:23
It is Reimann-integrable. Its just that no antiderivative of $x \in \mathbb{R} \mapsto \exp(x^2)$ can be written down in terms of certain operations applied to certain primitive functions. (Not really that profound, imo). By an antiderivative of $f : I \rightarrow \mathbb{R}$, I just mean a function $g : I \rightarrow \mathbb{R}$ such that $g' = f$. — goblin GONE, Jan 23 '14 at 09:25
@Teg : First question : "Can the function 1/x be integrated ?" Loosely answer from someone who doesn't known the function ln(x) : "No, it doesn't". Second question: "Can the function Exp(-x²) be integrated ?" Loosely answer from someone who doesn't known the function Erf(x) : "No, it doesn't". There are many questions and answers of this kind from people who doesn't know about special functions. The subject is discussed in the paper "Safari in the country of Special Functions" (overview for general public) : http://www.scribd.com/JJacquelin/documents — JJacquelin, Jan 23 '14 at 11:03
I'd say that this is not a duplicate of http://math.stackexchange.com/questions/523824/what-is-the-antiderivative-of-e-x2, since this question asks for a proof, and no proof was considered there. — Carsten S, Jan 23 '14 at 11:12

score 6 · Accepted Answer · answered Jan 23 '14 at 10:38

Your teacher probably said that $\exp(-x^2)$ (or $\exp(x^2)$) does not have an antiderivative that can be expressed using elementary functions:

In mathematics, an elementary function is a function of one variable built from a finite number of exponentials, logarithms, constants, and nth roots through composition and combinations using the four elementary operations (+ – × ÷).

The set of elementary functions forms a field of functions, equipped with a derivative operation: a differential field of functions. These kind of fields have been introduced by Liouville, in order to demonstrate his famous theorem which states that

the antiderivatives of certain elementary functions cannot themselves be expressed as elementary functions.

The proof uses algebra in the differential field of functions. It would be by all means not productive to reproduce it here, but there are some references you might want to check:

The modern version of the demonstration of this theorem is an open-access article by Rosenlicht.
A post by Matthew Wiener provides several examples.

You can't usub using $u = x^2$? Then $du = 2xdx$ oh wait nevermind... — Jwan622, Feb 23 '19 at 16:16

score 0 · Answer 2 · edited Oct 02 '24 at 07:21

As V. Rossetto wrote, your teacher more than likely told that the antiderivative of such a function cannot be obtained on the basis of simple functions.

For this integrand, the antiderivative is $\frac{\sqrt{\pi}}2\text{Erfi}(x)$, where $\text{Erfi}(x)$ is the imaginary error function $\frac{\text{Erf}(iz)}{i}$. The error function $\text{Erf}(x)$ is the integral of the Gaussian distribution, given by $$\text{Erf}(z)= \frac2{\sqrt{\pi}} \int_0^z e^{-t^2}\mathrm dt$$

I hope that you better percieve the vicious circle we move around.

Why can't $e^{x^2}$ be integrated

2 Answers2