Closed Form SolutionsDate: 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? Sincerely, 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 Check out our web site! http://mathforum.org/dr.math/ |
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