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Math Help - Notation for the nth Integral

  1. #1
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    Notation for the nth Integral

    If we want to represent the nth derivative of (ax+b)^m, then out of many we can write:

    \displaystyle f^{(n)}(x) = a^n(ax+b)^{m-n}\prod_{n=0}^{n-1}(m-n) (Lagrange's notation).


    \displaystyle \frac{d^ny}{dx^n} = a^n(ax+b)^{m-n}\prod_{n=0}^{n-1}(m-n) (Leibniz's notation).

    But what notation do we use for representing the reverse — that is, for the nth integral of the RHS?
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  2. #2
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    Is there in anything wrong/ambiguous about using \int f(x)\;{dx_{n}} for the nth integral of f(x)?


    [Got that idea from wikipedia article that says the multiple integration of a function in n variables,
    f(x_1, x_2, ..., x_n) , over a domain D is represented by \int \cdots \int_\mathbf{D}\;f(x_1,x_2,\ldots,x_n) \;\mathbf{d}x_1 \!\cdots\mathbf{d}x_n]
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  3. #3
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    You will also sometimes see D^n(f) for the nth derivative and D^{-n}(f) for the nth anti-derivative. Of course, D^0(f)= f. If you use f^{(n)}(x) for the nth derivative, you could use f^{(-n)}(x) for the nth anti-derivative.

    TheCoffeeMachine, wouldn't the form you give be likely to be confused with the integral with respect to the nth coordinate variable?
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  4. #4
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    Quote Originally Posted by TheCoffeeMachine View Post
    Is there in anything wrong/ambiguous about using \int f(x)\;{dx_{n}} for the nth integral of f(x)?


    [Got that idea from wikipedia article that says the multiple integration of a function in n variables,
    f(x_1, x_2, ..., x_n) , over a domain D is represented by \int \cdots \int_\mathbf{D}\;f(x_1,x_2,\ldots,x_n) \;\mathbf{d}x_1 \!\cdots\mathbf{d}x_n]
    Yes, because what you have written refers to integrating a vector function with respect to the nth component.
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