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Is there any analytic method to compute $x$ such that $e^{-x^{2}} = x$? Came across this in some game theory I am doing.

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up vote 7 down vote accepted

Rewrite it as $xe^{x^2}=1$. Square and then double both sides (in that order) to obtain


Now apply the Lambert-W function to both sides and get $2x^2=W(2)$, hence

$$x=\sqrt{\frac{W(2)}{2}}\approx 0.652919.$$

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