> Unfortunately, there’s a catch. Ullisch’s solution ... can’t tell you, in a practical sense, how long to make the goat’s leash. Approximations are still required...
Ok, then in what sense is this an exact solution?
Ok, then in what sense is this an exact solution?
If so, as this example shows (contour integrals in a closed form?) “closed form” is loosely defined, but I think most would say something with an infinite sum wouldn’t be one (but then, https://en.wikipedia.org/wiki/Closed-form_expression#Analyti... says:
“In particular, special functions such as the Bessel functions and the gamma function are usually allowed, and often so are infinite series and continued fractions. On the other hand, limits in general, and integrals in particular, are typically excluded.[citation needed]”
However, it seems the problem is hard for other reasons. I had assumed, without checking, that they'd put the center of the circle at 3pi/8 because it's a pole. Nope! As best I can tell from some graphing tools, there is indeed precisely one pole in the circle, and it's on the real line, but it's close to the right endpoint; I don't know that it has any nice form. So I imagine that getting any sort of exact series expansion around there -- or even just getting exactly the first few terms, i.e. the first few derivatives there, which would be all you'd need -- would be difficult for that reason.
(Although, the higher the order of the zero, the more initial terms you'd need...)
Edit: Actually, I guess it looks like a zero of order 1? Except that doesn't make sense, because then the top contour integral would be zero...
This is like those articles that are about a picture but don't include the picture.
As the article says, the solution to the goat problem isn't practically constructible:
"It’s a bit more abstruse — the ratio of two so-called contour integral expressions, with numerous trigonometric terms thrown into the mix"
That's like saying that the goat-grass system as described is exact, and no approximation is needed. I can write 1 x sqrt(2) just as easily as I can write 1 x (goat-grass constant). We arbitrarily choose what constants and symbols are allowed when we use the phrase "exact solution." Philosophically, every solution is at most breaking down an answer into other solutions.
Interestingly, that's as far as it goes; if you start with a line of length 1, compass and straightedge will let you construct a line with length equal to the square root of any rational number. You can do addition, multiplication, subtraction, division... and square roots, and there things come to a stop.
How? How did you get a string of the correct length?
Note that the tool you described is a compass.
But those are not symbols. We're talking symbols (digits).
> you can't write it down as a decimal
sin(cos(log(98234/123)+tan(exp(pi/5))-1)+pi + integral(complicated function(x) dx from 0 to 1)
It doesn't tell you in a practical sense how long to make the goat's leash. So you find the numerical version to however many decimals you want to get a useful approximation.
r=very-difficult-to-compute-expression