Closed form solution

koolraj09

Hi Guys,
What does the term Closed form solution mean?

Mentallic

It means the expression is not infinite in length or can be expressed in a much shorter way.
For example, for each of the following equalities, the right side is a closed form expression while the left isn't.

[tex]1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...=e^x[/tex]

[tex]1+2+3+...+n=\frac{n(n+1)}{2}[/tex]

[tex]1+\frac{1}{1+\frac{1}{1+\frac{1}{...}}}=\frac{1+\sqrt{5}}{2}[/tex]

DarthPickley

I think it also means you can't have an integral or summation symbol of any sort, you have to have simple functions and algebraic operations and numbers. also, what would be a closed form of the roots of x^5 - x + 1 = 0 ? you can't get it by infinite process unless you define a recursive function (such as newton's method) that converges to it, so what is an open form of it?

vikas.tyagi5

If the solution is exact, its a closed form solution. For example, a polynomial of fourth order can be solved using quadratic formulae as base(though tedious) or using simple approximate numerical methods. If u solve for exact values(using quadratic formulae), its a closed form solution. So its obvious it can not contain any infinity terms too.

JJacquelin

The roots of the quintic equation can be expressed on closed form, thanks to the theta functions which are related to the Jacobi elliptic functions. (Very arduous)
http://mathworld.wolfram.com/QuinticEquation.html

pmsrw3

But this illustrates one of the difficulties in answering the question. The phrase "closed-form expression" is ambiguous -- it depends on what you define as allowable elementary functions. I mean, if I'm allowed to define new functions, I can express anything in closed form -- I just define a function whose value is the solution to my problem. This sounds like cheating, but that's basically where most of our more recondite special functions come from: elliptic functions, Bessel function, Hypergeometric functions, etc. There was no closed-form expression to the problem so some old 19th century German said, "I hereby define this function to be the solution." And presto! the problem had a closed-form solution.

JJacquelin

I agree, but even if this way to see things is correct it is still incomplete. Of course, it should be too easy to define a new special function as the solution of a problem and then, to say : The problem as a solution which is expressed thanks to the new special function !
Special functions are more than that. For exemple, consider the Riemann zeta function.
If you say : I define a closed-form expression f(x) for the infinite series of general term 1/n^x (for n=1 to infinity), the function f(x) is defined only for x>1 since the series doesn't converges for x<1. However zeta(x) is defied for any real x (except x=1) and even for complex x. The special function zeta covers much more background : integral definition of the function, analytic continuation and much more.
In fact, when we use a closed-form expression to express the solution of a problem, we refer to a background of knowledge and we give a relationship to standard functions, i.e. functions which have been widely studied.
If the solution of a problem is only used to define à new special function, a relationship is not established to any previous background. So, this supposedly "closed-form" is useless.
A funny example is given as a preamble in the paper "Sophomores Dream Function" (pp.2-3). By the link :
http://www.scribd.com/JJacquelin/documents
This is also a main theme in the paper "Safari au pays des fonctions speciales" ( not translated yet), same link.

pmsrw3

Yes -- I was exaggerating a bit for effect. Actually, elliptic functions are an interesting case, because, although it is true that Jacobi developed them essentially as I described, as an easy-out function to solve a particular class of problems, they're more broadly applicable. I doubt he was thinking of solving quintics.