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A basic question: What does closed form mean?

  1. Jan 20, 2010 #1
    A basic question: What does "closed form" mean?

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

    What does closed form mean?
     
  2. jcsd
  3. Jan 20, 2010 #2
    Re: A basic question: What does "closed form" mean?

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

    [itex]a+ ax + ax^2 + ax^3 + ... = \frac{a}{1-x}[/itex]

    the right-hand side is a closed form of the left-hand side
     
  4. Jan 20, 2010 #3

    disregardthat

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

    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."
     
  5. Jan 21, 2010 #4
    Re: A basic question: What does "closed form" mean?

    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
     
    Last edited: Jan 21, 2010
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