Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 68 / 1 / a

For real $x$, the function $F(x)=x^2{e}^{-x}$ has derivative $F^{\prime }(x)=x(2-x)e^{-x}$. It decreases from infinity to zero on the negative half-line, increases from zero to $4e^{-2}$ on $(0,2)$, and then decreases to zero. Hence sufficiently small positive $ϵ$ gives exactly three real roots: two near zero and one tending to positive infinity.

Set $\delta =\sqrt{ϵ}$ and write either small root as $x=s\delta +A{\delta }^2+O({\delta }^3)$, with $s=\pm 1$. Expanding $x^2{e}^{-x}$ gives ${\delta }^2+s(2A-1){\delta }^3+O({\delta }^4)$. Matching its value to ${\delta }^2$ fixes $A=1/2$. Thus the two asymptotic expansions areFor the large root, put $L=\log (1/ϵ)$. Taking logarithms gives $x-2\log x=L$, so $x/L\to 1$. Writing $x=L+r$ then gives $r=2\log L+2\log (1+r/L)$. Iteration yieldsIts two asymptotic expansion terms are of different logarithmic orders; a regular power series in $ϵ$ cannot describe it.

As an exact check, the real branches of Lambert W give $x_{-}=-2W_0(\sqrt{ϵ}/2)$, $x_{+}=-2W_0(-\sqrt{ϵ}/2)$ and $x_{\mathrm{large}}=-2W_{-1}(-\sqrt{ϵ}/2)$. If complex roots are intended instead, there are infinitely many, given by $-2W_j(\pm \sqrt{ϵ}/2)$. For each fixed nonprincipal branch their two logarithmic terms follow from $W_j(z)=L_j-\log L_j+O(\log |L_j|/|L_j|)$, where $L_j=\log z+2\pi ij$ and logarithms are continued on that branch. The three real roots are the ones selected above.