Why there is no f(x) with d(f(x))/dx = exp(-x^2)?

Question

OK Puggy, but I want to distinguish between:
- Nobody found jet this expression
- Nobody will ever find this expression because it's impossible, there is no solution.
And I want to know why it is not possible to find f(x)

Answer 1

Look at this Wolfram site where it explains the Erf function which is part of the solution to this equation. The solution is:

f(x) = (1/2)(√π) Erf(x)

In mathematics, we have the "elementary functions", which are those taught in high school, like Sin(x), e^x, Log(x), etc. But professionally speaking, there's a vast range of "special functions" which usually arises from solutions to equations such as this one, and they're an important part of the repertoire of mathematicians, physicists, and engineers who have to work with such equations all the time. Have a look at another Wolfram site that lists a good number of these functions, next link.

There's only so much one can do with the "elementary functions" alone.

Answer 2

This integral leads to a thing called the 'error function' (usually written erf(x)) which, as Puggy pointed out, cannot be described in terms of the elementary functions.

Just type 'error function' into any search engine and there are probably hundreds of sites that will discuss it in detail.


Doug

Answer 3

Actually, there exists an f(x); it just cannot be expressed in terms of elementary functions.

Integral ( e^(-x^2) dx ) would be the answer to f(x). But no method of integrating that we know of will work (such as substitution, integration by parts, trigonometric identities, trigonometric substitution, partial fractions ...)