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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?

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/
```
Associated Topics:
High School Functions

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