This is motivated by this question and the fact that I have no access to Timothy Chow's paper What Is a Closed-Form Number? indicated there by
Qiaochu Yuan.
If an equation $f(x)=0$ has no closed form solution, what does it normally
mean? Added: $f$ may depend (and normally does) on parameters.
To me this is equivalent to say that one cannot solve it for $x$ in
the sense that there is no elementary expression $g(c_{1},c_{2},\ldots
,c_{p})$ consisting only of a finite number of polynomials, rational
functions, roots, exponentials, logarithmic and trigonometric functions,
absolute values, integer and fractional parts, such that
$f(g(c_{1},c_{2},\ldots ,c_{p}))=0$.
Answer
I would say it very much depends on the context, and what tools are at your disposal. For instance, telling a student who's just mastered the usual tricks of integrating elementary functions that
$$\int\frac{\exp{u}-1}{u}\mathrm{d}u$$
and
$$\int\sqrt{(u+1)(u^2+1)}\mathrm{d}u$$
have no closed form solutions is just the fancy way of saying "no, you can't do these integrals yet; you don't have the tools". To a working scientist who uses exponential and elliptic integrals, however, they do have closed forms.
In a similar vein, when we say that nonlinear equations, whether algebraic ones like $x^5-x+1=0$ or transcendental ones like $\frac{\pi}{4}=v-\frac{\sin\;v}{2}$ have no closed form solutions, what we're really saying is that we can't represent solutions to these in terms of functions that we know (and love?). (For the first one, though, if you know hypergeometric or theta functions, then yes, it has a closed form.)
I believe it is fair to say that for as long as we haven't seen the solution to an integral, sum, product, continued fraction, differential equation, or nonlinear equation frequently enough in applications to give it a standard name and notation, we just cop out and say "nope, it doesn't have a closed form".
No comments:
Post a Comment