Let's start with the Cauchy Formula for Repeated Integration.

Given a function f, a lower value a, and a value n representing the number of integrations we want to do, we can calculate the value of f

^{ -n}using a single integral of the following form:

Note that in order to correctly substitute x with the dummy variable t in Sage, I needed to use a dictionary subs call. Many thanks to the sage-support google group for their assistance with working this out.

It should be clear, that this integral multiplies the function f with appropriate powers of x, integrates over a dummy variable, and then divides by the correct constant.

The cool part comes with Reimann and Liouville, who realized that if you replace the factorial with a gamma function, there is no reason why the value of n couldn't be fractional (or irrational or even complex).

Thus, the 1/2 integral of x from a=0 is given below. Not surprising that it's a power of 3/2.

Note that the second line 1/2 integrates again to get the familiar x^2/2.

Now the cool part is that by using such an integral, we can then use it to define partial differentials. For example if we wanted the 1/2 differential of x from a=0, we would 1/2 integrate, then differentiate using normal calculus. Thus the 1/2 differential of x is:

Note that we got as suspected, x to the power of 1/2.

Thus, it is possible to differentiate or integrate continuously over fractional powers of n.

It gets really interesting once you realize that the Reimann-Liouville integral is also stable for complex numbers n, as long as the real part of n is larger than zero.

Thus, to finish, with a little fiddling, I show you the imaginary integral of x:

Notice that I integrated to (1 + i) to ensure the real part of n was larger than zero, then differentiated once. Also note that the e^(i log(x)) terms should simplify to x^i.

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