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Thread: 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 $\displaystyle (ax+b)^m$, then out of many we can write:

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


    $\displaystyle \displaystyle \frac{d^ny}{dx^n} = a^n(ax+b)^{m-n}\prod_{n=0}^{n-1}(m-n)$ (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 $\displaystyle \int f(x)\;{dx_{n}}$ for the nth integral of $\displaystyle f(x)$?


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


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