1

Is it possible to obtain the solution of

$$e^{\sqrt{x^{2} - x - 1}} = |x|$$

in closed form?

I know that $x$ must be somewhere between $\displaystyle\frac{\sqrt{5} + 1}{2}$ and $2$ after trying some substitutions. WolframAlpha gave me $x \approx 1.75036$. But apart from that I really have no idea how to start solving this.

Petch Puttichai
  • 637
  • 2
    not in closed form. – tired Jun 26 '15 at 16:45
  • Do you have a way to determine whether an equation can be solved in closed form? For this equation can you tell me how do you know that? – Petch Puttichai Jun 26 '15 at 16:52
  • i would guess that, because i know that even the much simpler looking equation $x=e^x$ has no solution in terms of elementary functions – tired Jun 26 '15 at 16:58
  • All i can get is, $e^t=\frac{1+\sqrt{5+(2t)^2}}{2}$ where $t=\sqrt{x^2-x-1}$ – Someone Jun 26 '15 at 17:21
  • Define $Q_r(e)$ to be the solution to this equation, if you derive enough identities and relations about the function, maybe this solution will be taken seriously one day ;) – Zach466920 Jun 26 '15 at 18:54

0 Answers0