Iterated integral

Iterated integral of function \(f\) is function \(J^n f\) expressed with iteration of integration:
 * \(\!\!\!\!\!\!\!\!\! (0) ~ ~ ~ \displaystyle

{J^n f(x)=} {\underbrace{\int_0^x \mathrm{d}x_1 \int_0^{x_1} \mathrm{d}x_2  .. \int_0^{x_n}\mathrm{d}x_n } ~ f(x_n)} \atop{{\phantom{I^n f(x)=}} n \rm ~ integrations} \)

The simple case of natural \(n\) can be generalized for integer rational and even complex values of \(n\); such a generalization is allows to deal with derivatives of arbitrary (negative, fractal and even complex order. The historical overview of the problem is available, the results about the fractional derivatives appear since year 1819.

Cauchi representation for the iterated integration
By the induction, the Cauchi formula for the nested integration can be proven


 * \(\displaystyle \!\!\!\!\!\!\ (1) ~ ~ ~ J_a^n f (x) = \frac{1}{(n\!-\!1)!} \int_a^x\left(x\!-\!t\right)^{n-1} f(t)\,\mathrm{d} t \)

at least for natural number \(n\). However, there are seductions to use such an expression also for complex values of \(n\), generalizing the nested integration and the differentiation for the non–integer number of iterations.

For \(a\!=\!0\), the lower subscript of \(J\) can be omitted; then it is assumed that \(J_a\)=\(J\). In the expression \(J_a^n f (x)\), it is assumed that first, the operation \(J_a\) is defined, then it is applied \(n\) times to the function \(f\) and them the resulting function is evaluated qith value of its argument equal to \(x\), id est, \(J_a^n f (x)=(((J_a)^n) f) (x)\). Some ambiguity appears because often the symbol of multiplication, the "centered dot" \(\cdot\) or "times" \(\times\) is omitted and the last expression, in principle, could be interpreted as multiplication of something to \(x\). However, the precedent expression is not a number, and in all cases when the confusion is expected, the multiplication signs should be recovered.

Other notations
Sometimes the letter \(I\) is used instead of \(J\), but that notation conflicts with notation \(I\) as imaginary unit (id est, \(\sqrt{-\!1}~\) ) in Mathematica and the Identity function, which is also denoted with \(I\). For this reason, in TORI, letter \(J\) is used, and, \(~J^0 \!=\! I~\) is denoted as Identity operator.

Also, sometimes, the some arbitrary constant \(a\) is used instead of \(0\) as the lower limit of integration. Such a generalization is equivalent of displacement of the argument of the integrand, id est, the argument of the function \(f\), for a constant \(a\).

Negative number of iterations
The expression (1) is valid for natural values of \(n\), and the iterated integration satisfies the group property; for natural numbers \(m\) and \(n\), the relation below holds:
 * \(\!\!\!\!\!\!\!\!\! (2)\displaystyle ~ ~ ~ J^m J^n = J^{m+n}\)

The generalization to the negative values of the number of integrations corresponds to the differentiation operator;
 * \(\!\!\!\!\!\!\!\!\! (3)\displaystyle ~ ~ ~

D^n f(x)=\frac{\mathrm d^n}{\mathrm d x^n} f(x)=J^{-n}f(x) \) For positive \(n,m\), the relation below holds:
 * \(\!\!\!\!\!\!\!\!\! (4)\displaystyle ~ ~ ~

D^n J^m f=J^{m-n}f \) However, the inverse relation \( ~ J^mD^n f=J^{m-n}f  ~ \) may fail; in particular, it fails while \(f\) is polynomial of order smaller than \(n\) and \(D^n f =0\)

Fractional integration and fractional derivatives
Expression (1) could be used for complex (and, in particular, fractional) values of \(n\), under condition that the integral converges. In this case, the expression \((x\!-\!t)^{n-1}\) is interpreted as \(\exp\big(\ln(x\!-\!t) (n\!-\!1) \big)\). Due to the cut line of the logarithm, the \(J^n f\) may have singularities and cutlines even if \(f\) is holomorphic function.

The generalization with lower limit of integration \(a\ne 0\) seem to have not so much sense. However, this parameter becomes important, if for some proof one needs to differentiate or integrate with respect to this parameter.

Questions
One may consider application of the fractional differentiation to some specific functions; for example, the polynomial or the exponential. Site Maa. suggests several questions about the fractional derivatives:

Q1 In this case does \(D^\alpha (u \mathrm{e}^{p x} + v \mathrm{e}^{q x} = u D^\alpha \mathrm{e}^{p x} + v D^\alpha \mathrm{e}^{q x} ~ \)?

Q2	In this case does \(D^\alpha D^\beta \mathrm{e}^{px} = D^{\alpha+\beta} \mathrm{e}^{px} ~\)?

Q3 Is \(~ D^{-1} \mathrm {e}^{px}=\int_0^x \mathrm{e}^{pz} \mathrm d z ~\) always true, or is there something missing?

Q4 What general class of functions could be differentiated fractionally be means of the idea contained in (1)? Q1.
 * \(\!\!\!\!\!\!\!\!\! (4)\displaystyle ~ ~ ~

D^n J^m f=J^{m-n}f \)

Caputo differentiation
The generalization to the non–integer number of iteration should be solution of the equation (2) for non–integer \(n\). Such a generalization is attributed to Michele Caputo . The construction of the solution is based on the following formula:
 * \(\!\!\!\!\!\!\!\!\! (5)\displaystyle ~ ~ ~ \frac{1}{\mathrm{Factorial}(b)} \int_0^x t^p (x - t)^b ~\mathrm{d}t =

\frac{x^{1+b+p} ~ \mathrm{Factorial}(p)} {\mathrm{Factorial}(1\!+\!b\!+\!p)} \) valid at least for real values of \(p\) and \(b\) such that \(p\!>\!-1\) and \(b\!>\!-1\). For the generalization of (1) to the case of real \(n\), it is sufficient to express \(J^n\) for \(0<n<1\). At least for function \(f\) that can be represented with series
 * \(\!\!\!\!\!\!\! (6)\displaystyle ~ ~ ~

f(x)=\sum_{k=0}^\infty c_k x^{p+k}\) for some constants \(p\), and \(p\), the iterated integration can be defined as follows:
 * \(\!\!\!\!\!\!\!(7)\displaystyle ~ ~ ~

J^n f (x)= \int_0^x \mathrm{d}t~ f(t) (x-t)^n\)

This some other useful expressions, this formula is attributed to Cauchi.

As the differentiation and integration, the iterated integration is linear operation; at least for real \(n>-1\), for any complex coefficients \(\alpha\) and \(\beta\) and functions \(u\) an \(v\) representable in form (6), and \(g(z)=\alpha u(z) + \beta v(z)\) the following relation holds:
 * \(\!\!\!\!\!\!\! (8)\displaystyle ~ ~ ~

J^n g (x)= \alpha J^n u(x)+ \beta J^n v(x) \).

For non–integer \(n\), the iterated integration \(J^n\) by (7) is called the Caputo integration, after professor Michele Caputo; the inverse operation is called '''the Caputo differentiation. Such operations seem to be useful for simplification of physical problems, although the mathematical articles rarely indicate the possible application of the formalism of the fractional integration and derivatives, if at al.