อ้างอิง:
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Problem. Solve the equation $$x+\mathrm{e}^x=0\,.$$
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Let $x\in\mathbb{C}$ be such that $x+\exp(x)=0$. Observe that
$$(-x)\,\exp(-x)=\exp(x)\,\exp(-x)=1\,.$$
Therefore, $z:=-x$ is a solution to
$$z\,\exp(z)=1\,.$$
For each $t\in\mathbb{C}$, we know that all solutions to $z\,\exp(z)=t$ are of the form
$$z=W_k(t)\,,$$
where $W_k$ is the $k$-th branch of the Lambert $W$-function for each $k\in\mathbb{Z}$. Therefore, all solutions to the required equation are of the form
$$x=-z=-W_k(1)\,,$$
where $k\in\mathbb{Z}$. Here are some values of $-W_k(1)$:
$$-W_{-2}(1)\approx 2.401585+10.776300\,\mathrm{i}\,,$$
$$-W_{-1}(1)\approx 1.533813+4.375185\,\mathrm{i}\,,$$
$$-W_0(1)\approx -0.5671533\,,$$
$$-W_{+1}(1)\approx 1.533813-4.375185\,\mathrm{i}\,,$$
and
$$-W_{+2}(1)\approx 2.401585-10.776300\,\mathrm{i}\,.$$
Note that $x=-W_0(1)$ is the only real solution to this equation.