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Comment by youareawesome | Hacker News Reader

This is easily solved using the Lambert W function:

Observe: W(x * e ^ x) = x

    x ^ x = y
    ln(x ^ x) = ln(y)
    ln(x) * x = ln(y)
    ln(x) * e ^ ln(x) = ln(y)
    W(ln(x) * e ^ ln(x)) = W(ln(y))
    ln(x) = W(ln(y))
    x = e^W(ln(y))

https://en.wikipedia.org/wiki/Lambert_W_function

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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.

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

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.

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