## Monday Math 24: Repeated Integrals

Consider an antiderivative of the function , such as . This can be written as a linear operator (and is known as the indefinite integral operator, or Volterra operator), which we will write here as J: so that . (Note that for the rest of this post, we assume  is sufficiently continuous and integrable on the regions in question).

Now, let us consider repeated applications of the operator. For example,
. We write the operator of J applied twice as . Similarly, we have  given by , and so on for , n any positive integer. Is there a way to find the n-th antiderivative  without needing to perform n integrations?

The answer is yes. We can reduce it to a single integral. Observe in the case of :
.
Treating this as a double integral, and considering the region of the tu plane over which it is integrated, we can reverse the order of integration:
.
As  is a constant with respect to t, we find that the inner integral is simply , and we have:


Similarly, we have
,
which with reordering to make integration by u3 last, and then performing the u1 and u2 integrals, we obtain (renaming u3 as u):
.

If we were to continue, we would find a pattern:


and so on, giving us the formula:
,
which is known as the Cauchy formula for repeated integration.

We can use mathematical induction to prove that this formula is true for all positive integer n. First, the n=1 case: we see it gives:
,
and so the formula holds for n=1.
Since , by definition, we see that if we take the derivative of both sides with respect to x,
.
Now, we can take the derivative of our formula with respect to x, using the variable limit form of the Leibniz integral rule:
,
which is our formula’s value for . Thus our formula holds for  given , and so it holds for all integer n>0 by induction.

Now, we see that the integral  can be performed for n>0 even if n is not an integer. Recall, however, that for positive integer n, , where  is the gamma function. Thus, if we replace the factorial in our formula with the corresponding gamma function, we obtain the formula
,
which can be computed for n a positive real number. This is the basis of the Riemann-Liouville differintegral, the most often used differintegral (operator combining differentiation and integration) in fractional calculus.