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Consider equation $\exp(-x) = x$.

Solution is so-called $\text{LambertW}(1)$.

It is beautiful special function called LambertW function or polylog.

Now consider $\cos(x) = x$ and $\sin(x) = x$.

What beautiful special functions solve above equations?

Nigel1
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    Presumably you mean ProductLog or similar, not polylog(arithm), which is something else: https://en.wikipedia.org/wiki/Polylogarithm – Travis Willse Jun 15 '17 at 14:25
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    Also, $\exp x = x$ has no real solutions. The quantity $\textrm{LambertW}(1)$ is the solution to $\exp(-x) = x$. – Travis Willse Jun 15 '17 at 14:31
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    I've voted to close as a duplicate of the indicated question. It doesn't address the equation $\sin x = x$, but that case is almost trivial: It's not hard to show that $x = 0$ is the only real solution, and obviously no special functions are required to express that value. – Travis Willse Jun 15 '17 at 14:33

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