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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: $$\int_{g_2(x_1,x_2,\dots,x_n)}^{f_2(x_1,x_2,\dots,x_n)}{\left(\int_{g_1(x_1,x_2,\dots,x_n)}^{f_1(x_1,x_2,\dots,x_m)}{f(x_1,x_2,\dots,x_m)dx_i}\right)dx_j}$$

Where or how can I find more about this second form of nesting?

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](http://en.wikipedia.org/wiki/Integral_transform). No "nesting" is necessary. As for "nested integrals" with functions as limits, look up information about [iterated integrals](http://en.wikipedia.org/wiki/Iterated_integral) in any multivariable calculus text.2012-07-07
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    @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.2012-07-07
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    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...)2012-07-07
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    @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".2012-07-07

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