chalkdust expression would work smoothly… except for that dodgy factorial! The value of n factorial (wrien as n!, possibly the worst symbol ever used in maths since now we cannot express a number with surprise) is the product of the numbers from 1 through to n, ie 1 × 2 × · · · × n. What, then, would the factorial of 9¾ be? Maybe close to 10! but not quite there yet? Luckily for us, an expression for the factorial of a real number has intrigued mathematicians for centuries, and brilliant minds like Euler and Gauss, amongst others, have worked on this issue. They defined the gamma function, Γ, in such a way that it has the two properties we need: first, Γ(n) = (n − 1)!, so we can use Γ(n) instead of the factorial. Second—and even more importantly, given that we are dealing with fractions here—is that the function is well-defined and continuous for every positive real number, so we can now compute the factorial of 9¾, which is only 57% of the value of 10!. Now, we can write the repeated integral as 1 D f (x) = Γ(ν) −ν
(x − t)ν−1 f (t) dt,
which gives the same result as before when ν is a positive integer and is well-defined when ν is not an integer. The integral above is known as the fractional integral of the function f. Awesome!
Graph of the gamma function Γ(x).
What happens if we take the derivative of the repeated integral? Easy! We get the fractional derivative, right? Not quite, since we have two options: we could either diﬀerentiate the original function first and then take the fractional integral, or we could fractionally integrate first and then take the derivative. Damn! Both definitions are equally valid and we mathematicians hate having two definitions for the same thing. But are they even the same thing? If we diﬀerentiate first and then take the fractional integral—known as the Caputo derivative—we don’t necessarily get the same result as if we fractionally integrate a function first and then take its derivative. The laer is called the Riemann–Liouville derivative or simply the fractional derivative since it is the one more frequently used. As an example, let’s look at the 9¾ derivative of a polynomial, say f (x) = x 9 . The 9¾ Caputo derivative is zero, since we first diﬀerentiate x 9 ten times, which is zero, and then integrate it; but chalkdustmagazine.com