It's my understanding that what is and isn't "closed form" is rather arbitrary. Functions which are used frequently – like exp() – are elevated to closed form status, and yet you can't evaluate exp() in a finite number of steps. So how is the explicit solution to the goat problem objectively different?
Yes I also feel that way to some degree, but I just never considered an integral to be closed form. There is some argument to be made for exp, as it is considered an 'elementary function'. I was going off this statement from wikipedia on closed form expressions
"It may contain constants, variables, certain "well-known" operations (e.g., + − × ÷), and functions (e.g., nth root, exponent, logarithm, trigonometric functions, and inverse hyperbolic functions), but usually no limit, differentiation, or integration."
Edit: I think if you say an integral is closed form, you must also admit that a limit is closed form, since an integral is defined in terms of limits (though technically more restrictive). In that case, you should also admit that we already had a closed form expression for this number, as it could be expressed as a limit of an iterative process.
> There is some argument to be made for exp, as it is considered an 'elementary function'.
But exp() is defined as the limit over an infinite sum, so why does it get to be an elementary function?
My point it that the distinction between closed form and non-closed form is arbitrary, and that there is no qualitative difference. In fact, limit and integral are just (higher-order) functions as well – and rather ubiquitous ones, so why aren't they considered elemental?
You're not wrong, in that closed-form is not a clearly-defined term.
However, mathematicians rarely do numerical calculations, so exp(), gamma, etc. are considered elementary functions, and there's no urgency to translate from symbols to numbers.
Physicists who do calculations might be more restrictive about what closed-form means if they intend to compute a numeric result.
Source: I studied pure mathematics, and we never used calculators - it was laughable, in fact. Most of our series never converged in the first place. :)
But it is exact.
> Not really [a] closed form solution
It's my understanding that what is and isn't "closed form" is rather arbitrary. Functions which are used frequently – like exp() – are elevated to closed form status, and yet you can't evaluate exp() in a finite number of steps. So how is the explicit solution to the goat problem objectively different?