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Discrete Calculus

  1. Nov 30, 2004 #1
    Is there such thing as discrete calculus? Or are there general rules to find derivatives and integrals of functions whose domains are restricted to integers or some other discrete values?
     
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  3. Nov 30, 2004 #2

    Hurkyl

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    An integral over a discrete set is simply a sum! In the general case, the integral of a function [itex]f[/itex] with respect to a measure [itex]\mu[/itex] can be computed by:

    [tex]
    \int_A f d\mu = \sum_{a \in A} f(a) \mu(a)
    [/tex]





    There is a discrete analog of a derivative called a difference:

    [tex]
    \Delta_hf(x) = f(x + h) - f(x)
    [/tex]

    (when h is omitted, assume it's 1)

    And difference equations have many similarities with differential equations. For example, one can "solve" for the Fibonacci sequence which is defined by a linear second-order homogenous difference equation:

    [tex]
    \Delta^2 F + \Delta F - F = 0 | F(0) = 0, F(1) = 1
    [/tex]

    whose solution technique is directly analogous to that of similar differential equations: (use [itex]F(r) = a^r[/itex] as a putative solution, get two linearly independent solutions, and take a linear combination that satisfies the initial conditions)

    There's a more general concept here called a skew derivation (or [itex]\sigma[/itex]-derivation) of which both the ordinary derivative and this finite difference are examples.


    And, of course, there's the antidifference operator, also called the summation operator, which bears a similar to indefinite integrals. For instance, you can even do summation by parts. :smile:
     
    Last edited: Nov 30, 2004
  4. Dec 1, 2004 #3

    HallsofIvy

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    It's more often called "finite differences" rather than "discrete calculus".

    Try a google search on "finite differences". Boole wrote a book on it that is still published by Dover.
     
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