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  "body": "\n## Summary\n\nThe Lambert W function is the [[inverse|0J2WAR83WP]] of the map $w \\mapsto w e^w$, so $W(x)\\, e^{W(x)} = x$.\nOver the complex numbers it is multivalued, with countably many branches $W_k$; on the reals it has two branches $W_0$ and $W_{-1}$.\nIt solves many equations that mix exponentials and polynomials, such as $x = e^{-x}$ or $a^x = x$.\nIt appears in combinatorics, in delay differential equations and in physics.\n\n## Explanations\n\n### L1 human\n\nThe function $w e^w$ is not inverted by logarithms or roots, so its inverse gets a name of its own.\nTo solve $x e^x = 1$, one writes $x = W(1) \\approx 0.567$.\nTo solve $x = e^{-x}$, one multiplies by $e^x$ to get $x e^x = 1$, so the answer is the same number.\nOn the real line, $w e^w$ has a minimum $-1/e$ at $w = -1$, so the equation $w e^w = x$ has two real solutions for $-1/e \u003c x \u003c 0$, and this is why there are two real branches.\n",
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