[throwlaplace]

That's pretty asinine since the W function is that inverse (basically). You're just saying "this is easily solved by easily solving".

Before anyone jumps on me: I'm perfectly familiar and comfortable with functions that aren't defined in closed form.

[knzhou]

That's true for almost any nontrivial transcendental algebraic or differential equation. Special functions to solve special cases are the norm.

We could repeat the same discussion at a lower level. Suppose you tell a bright middle schooler about the basics of integration, and the power rule. Then they ask,

"So what's the integral of 1/x? It can't be x^0/0..."

"That's a special case. It's a function called "natural logarithm", ln(x)."

"But what's the definition of that weird function?"

"It's defined to be the integral of 1/x." [0]

"That's pretty asinine..."

[0] https://en.wikipedia.org/wiki/Natural_logarithm#Definitions

[paulddraper]

To be fair, this is much of math.

The ratio of right triangle legs is "easily solved" by tan(x). But that's magical/cheating to someone who hasn't studied trigonometry.

Lambert-W is a well known function with known approximation methods; once you reduce your problem to it (in this case, trivially), you can lean on that knowledge from others.