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Closed Form Solutions

Date: 09/16/97 at 12:55:35
From: Scott Batterman
Subject: Closed form solutions

Dear Dr. Math,

What is the exact mathematical definition of a closed form solution?
Is a solution in "closed form" simply if an expression relating all of 
the variables can be derived for a problem solution, as opposed to 
some higer-level problems where there is either no solution, or the 
problem can only be solved incrementally or numerically?

Scott Batterman

Date: 09/22/97 at 13:12:27
From: Doctor Rob
Subject: Re: Closed form solutions

This is a very good question!  This matter has been debated by
mathematicians for some time, but without a good resolution.

Some formulas are agreed by all to be "in closed form."  Those are the
ones which contain only a finite number of symbols, and include only 
the operators +, -, *, /, and a small list of commonly occurring 
functions such as n-th roots, exponentials, logarithms, trigonometric 
functions, inverse trigonometric functions, greatest integer 
functions, factorials, and the like. 

More controversial would be formulas that include infinite summations 
or products, or more exotic functions, such as the Riemann zeta 
function, functions expressed as integrals of other functions that 
cannot be performed symbolically, functions that are solutions of 
differential equations (such as Bessel functions or hypergeometric 
functions), or some functions defined recursively. 

Some functions whose values are impossible to compute at some specific 
points would probably be agreed not to be in closed form (example:  
f(x) = 0 if x is an algebraic number, but f(x) = 1 if x is 
transcendental. For most numbers, we do not know if they are
transcendental or not).

I hope this is what you wanted.

-Doctor Rob,  The Math Forum
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Associated Topics:
High School Functions

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