# Defining an mathematical operator

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.

• Take a look at this question mathematica.stackexchange.com/questions/5030/… Aug 11 '20 at 12:54
• 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" Aug 12 '20 at 8:45