# Is there any body of knowledge or study of the fractional calculus on definite integrals?

The fractional calculus is partly about nested indefinite integrals. Is there any study or body of knowledge on nested DEFINITE integrals? For example, the fractional calculus helps with this integral: $$\int_0^{l_2}{\left(\int_0^{l_1}{f(l_1)dl_1}\right)dl_2}$$ It can be readily observed that these forms don't allow for any limits of integration other than a new variable to replace the previous variable of integration.

What I am looking for is nested integrals that have functions for the limits of integration: $$\int_{g_2(x_1,x_2,\dots,x_n)}^{f_2(x_1,x_2,\dots,x_n)}{\left(\int_{g_1(x_1,x_2,\dots,x_n)}^{f_1(x_1,x_2,\dots,x_m)}{f(x_1,x_2,\dots,x_m)dx_i}\right)dx_j}$$

REFINEMENT

The first expression above is combined into an operator, say $J$, to the second power in the fractional calculus. I'm looking for an extension of this so that the second expression can be combined into a similar operator, say $J_2$. I'm wondering where this has been done. It seems to be more than just iterated integrals.

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I don't quite follow how this connects back to fractional calculus. Fractional integrals can be defined as a type of integral transform. No "nesting" is necessary. As for "nested integrals" with functions as limits, look up information about iterated integrals in any multivariable calculus text. –  Bill Cook Jul 7 '12 at 1:14
@BillCook: I'm seeking to combine iterated integrals into a single integral or operator, if you will. The idea is to create a set of operators that perform various operations using the "power" of integrals to do the dirty work. I guess I'm looking to see what kind of work has been done that combines operator theory and iterated integrals. –  Matt Groff Jul 7 '12 at 1:39
You do realize that differintegrals in general necessarily require a lower limit as part of their complete representation? (The exception, of course, being nonnegative integer orders...) –  Guess who it is. Jul 7 '12 at 3:55
@J.M.: Is this lower limit used in every step of the integration, or just at the end? I'm hoping to find a way to use limits of integration at every "iteration". –  Matt Groff Jul 7 '12 at 4:12