3

$$\mathfrak{E} = \int \exp \Big[ \operatorname{erf}{\Big(\frac{x}{u} \Big)} \Big] \, dx $$

This integral has come up in my research and I have not found any known solutions (integral tables, wolfram alpha, matlab symbolic integration). I am still intrigued as the function has interesting characteristics that seem like it would be common in other physical systems. Plus, $\mathfrak{E}$ is continuously differentiable, which seems promising. But my math capabilities have not sufficed in pursuit of a solution. Has anyone out there dealt with this integral before or seen it come up?

Any answers or comments are much appreciated!

EDIT: Many thanks to Robert for pointing out my misuse of the theorem of Liouville (removed) and for Michael's help with my formatting!

AstroCamber
  • 31
  • If I am not mistaken, based on derivatives, the solution $\mathfrak{E}$ is also the solution to the equation,
    $$ \frac{d^2\mathfrak{E}}{dx^2} - G\frac{d\mathfrak{E}}{dx} = 0$$
    Where $G = \frac{2}{\sqrt{\pi}u} \exp{\big[-\frac{x^2}{u^2} \big] }$.     – AstroCamber May 17 '18 at 00:42
  • 2
    When you write \text{erf} instead of \operatorname{erf}, then you don't get proper spacing in things like $3\operatorname{erf} A$ or $3\operatorname{erf}(A),$ and instead you see $3\text{erf} A$ or $3\text{erf}(A).$ I included both of those examples so that you can see the context-dependent nature of the spacing afforded by \operatorname{}. If you need it 50 times in one posting, you can use \newcommand at the beginning so that you need only type \erf. In a LaTeX document, the \newcommand comes above the \begin{document}. $\qquad$ – Michael Hardy May 17 '18 at 00:46
  • 2
    The theorem of Liouville you referenced is about whether $f e^g$ has an elementary antiderivative, where $f$ and $g$ are rational functions. $\text{erf}$ is not rational, so that theorem is irrelevant here. – Robert Israel May 17 '18 at 01:18
  • Your scaling parameter $u$ can be rid of. –  May 17 '18 at 15:49

0 Answers0