A basic question: What does closed form mean?

[seeker101]

A basic question: What does "closed form" mean?

"The point here is that $\sigma$ algebras are difficult but $$\pi$$ systems are easy: one can often write down in closed form the general element of a $$\pi$$ system while the general element event of $\mathbf B \mathbf$ is impossibly complicated" - From the "Diffusions" text by Rogers and Williams.

What does closed form mean?

 

[fourier jr]

hard to describe... I guess it's a formula that has a finite nmber of symbols. i know for an infinite geometric series,

$a+ ax + ax^2 + ax^3 + ... = \frac{a}{1-x}$

the right-hand side is a closed form of the left-hand side

 

[disregardthat]

I suppose it depends on the context. If you have a sum of infinite terms, "closed form" would perhaps be an explicit formula of a few terms.

Wiki got something on it:

"In mathematics, an expression is said to be a closed-form expression if, and only if, it can be expressed analytically in terms of a bounded number of certain "well-known" functions. Typically, these well-known functions are defined to be elementary functions; so infinite series, limits, and continued fractions are not permitted.

Similarly, an equation or system of equations is said to have a closed-form solution if, and only if, at least one solution can be expressed as a closed-form expression."

 

[Mensanator]

I don't think it is limited to infinite series.

Say you wanted the sum of 5 powers of 9: 9^4 + 9^3 + 9^2 + 9^1 +9^0

Not so hard, but what if there were a hundred terms?

Note that the sum is 11111₉. Or 88888₉/8₉.

But 88888₉ is 9⁵-1. For n terms, the closed form would be

(9ⁿ-1)/8