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I'm looking for a correct English term for the following concept: $$ If(x)=\int_0^x f(t)dt \\ g(x) = I(If(x)) \\ h(x) = I^{n}f(x) $$

What should be the precise name of $h(x)$? So far I came up with:

  • n-th order repeated integral
  • n-th order integral
  • nth order integral
  • nth integral

Thank you for your help in advance

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2 Answers 2

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One could probably call it any of those terms, but the interesting thing is how far it simplifies. Thanks to Cauchy's repeated integral formula, we have

$$I^nf(x)=\frac1{(n-1)!}\int_0^x(x-t)^{n-1}f(t)\ dt$$

which is very helpful for deducing things like

$$\int_1^x\ln(t)\ dt=\int_1^x\int_1^t\frac1y\ dy\ dt=\frac1{1!}\int_1^x\frac{x-t}t\ dt=x\ln(x)-x+1$$

which is rather simple.

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    $\begingroup$ Simply Beautiful Answer! $\endgroup$
    – Konstantin
    Mar 14, 2017 at 15:17
  • $\begingroup$ No problem and thanks! $\endgroup$ Mar 14, 2017 at 15:25
  • $\begingroup$ Is the above formula still valid when n is other than positive integer? Thank you $\endgroup$ Jan 20, 2020 at 13:21
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    $\begingroup$ @blindProgrammer The "nth integral" loses it's meaning when n is not a positive integer. One can define such a thing using the above if one so wishes though. $\endgroup$ Jan 20, 2020 at 13:41
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    $\begingroup$ @PrasenjitDWakode You treat it as a limit when $x\to\infty$ like you would any improper integral. $\endgroup$ Sep 30, 2020 at 12:34
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$f^{(-n)}(t)$ means nth antiderivative (indefinite integral) and $f^{(n)}(t)$ means nth derivative of $f(t)$

Write the notations as power of f written in round bracket. Positive power stands for derivative while negative power stands for antiderivative.

When n=0 , it would mean the function itself. I have seen these notations in many places.

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