Closed Form

What does "closed form" -mean-?
It means an answer in terms of -numbers- or -functions- that we
already know.

In general, we should -not- expect questions to have answers that are
familiar functions, i.e., we should not expect "closed form"
solutions, for the obvious reason that there are -lots- of different
phenomena out there!

We should only expect to get a closed form solution when a question is
-similar- to existing ones; generally, when there is a familiar -symmetry-
or -geometric- structure.

Indeed, if a function (or number) is interesting enough, we can add it
to our "box o' functions".
Indeed, that's exactly what we did for the trig and exponential
functions -- they're not (expressable as) rational/algebraic functions,
but they're useful, so we just added 'em to our "basic list".
...and likewise for $e$ and $\pi$.

What we do is:
- calculate them (in assorted ways, to desired accuracy)
- express them in other ways (power series, say, if possible)
- study their -properties-
  (they're usually rather more interesting and curvy than trig/exp)
...and this is in fact the core of analysis!

Examples of non-closed form problems:
- $\int e^{x^2}$
- The Elliptic Integral:
  circumference of an ellipse of radii $a,b$
  (equivalently, the integral ...)
In fact, $\int e^{-x^2}$ is so useful that we add it to our list
(actually, the scaled $\frac{1}{\sqrt{2\pi}} \int e^{-x^2/2}$;
 this is the error function, ubiquitous in statistics).

Examples of probably non-closed form numbers
(but I don't know; some are not proved to be).
- the solution to $x = \cos x$
- Euler's constant
