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Symbolic Methodology

  1. Aug 5, 2005 #1

    Equation1:
    [tex]\frac{d^2}{dx^2} (x^n) = \frac{d}{dx} \left[ \frac{d}{dx} (x^n) \right] [/tex]

    The LHS for Equation1 is the symbolic condensed version for the RHS, however, what is the LHS symbolic condensed version for Equation2 RHS?

    Equation2:
    [tex]\text{???} = \int \left[ \int \left( x^n dx \right) \right] \; dx[/tex]
     
  2. jcsd
  3. Aug 6, 2005 #2
    [tex]\int dx \int \left( x^n dx \right)[/tex]
     
  4. Aug 6, 2005 #3


    Interesting, I have never seen that version before. I was expecting something as:
    [tex]\int \int x^n dx dx = \int \left[ \int \left( x^n dx \right) \right] \; dx[/tex]

    However, what if I wanted to demonstrate an equation that must be integrated 10 times or even 100 times? Surely there must be a shorthand version?
     
  5. Aug 6, 2005 #4

    Hurkyl

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    It's somewhat rare to see iterated indefinite integrals: generally you would specify bounds, even if it's something like:

    [tex]
    \int_0^x \int_0^t f(s) \, ds \, dt
    [/tex]

    I've often seen high dimensional integrals written something like:

    [tex]
    \iint \cdots \int f(x_1, \ldots \, x_n) \, dx_1 \, dx_2 \, \cdots \, dx_n
    [/tex]

    with some additional text indicating the region of integration... or instead written as a single integral whose dummy variable ranges over a high-dimensional space.


    Another option, which I suspect is the best one for you, is to define an integral operator. For example, you could define the operator I via:

    [tex](If)(x) := \int_0^x f(t) \, dt[/tex]

    and then you could indicate an iterated integral by [itex]I^nf[/itex].
     
    Last edited: Aug 6, 2005
  6. Aug 6, 2005 #5
    You can write [tex]D^{-2}f(x)[/tex] and/or [tex]D^{-2}(x^n)[/tex].
     
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