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I'm trying to learn mathematica to use in my thesis. I have basic problems with definitions, calculations etc. Let me ask a question.

I want to define anti derivative operator and derivative operator, then a new operator with composition of these operators. Compositions must be identity, so if I give a function into new operator it will turn out the function what I gave it.

To explain better, I will try to write these in mathematical sense;

Here $D^m$ denotes standard derivative operator $m$ times and anti-derivative operator $I_m$ defined with

$ \qquad \begin{cases} f(x), & m=0 \\ \int_0^{x} \dfrac{(x-y)^{m-1}}{(m-1)!}f(u)du, & m\geq 1 \end{cases} $

Now, if we composite these functions, $[D^m\circ I_m](f)(x)$ these will give us directly $f(x)$. I want to define these operators in Mathematica and see composition of operators.

Actually these problem is not my main problem, but the answers will shows me what I need, then I will try to learn myself again.

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  • $\begingroup$ Take a look at this question mathematica.stackexchange.com/questions/5030/… $\endgroup$
    – flinty
    Aug 11 '20 at 12:54
  • $\begingroup$ In the definition of $I_m$ second line, upper part of fraction is $(x-u)^{(m-1)}$. There is a wrong writing but I cant edit it since I get an error "you have code (says for definition of $I_m$) but you did not write in code tag" $\endgroup$
    – XepheroS
    Aug 12 '20 at 8:45

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