chalkdust way to represent derivatives and integrals at the same time. So if we have a real number ν, why not express the fractional derivative of e cx as Dν e cx = cν e cx ? With this new derivative we know that if ν is an integer then the fractional expression has the same result as the traditional version of the derivative or integral. That seems like an important thing, right? If we want to generalise something, then we cannot change what was already there. If the fractional derivative is a linear operator (ie if a is a constant then D1/2 af (x) = aD1/2 f (x)) , then we would also obtain that [ ] [ ] D1/2 D1/2 e cx = D1/2 c1/2 e cx = c1/2 D1/2 [e cx ] = ce cx , so half diﬀerentiating the half derivative gives us the same result as just applying the first derivative. In fact for this very first definition of a fractional derivative, we get that Dν [D μ e cx ] = Dν+μ e cx = cν+μ e cx for all real values of ν and μ. Great: our fractional derivative has at least some properties that sound like necessary things. Diﬀerentiating a derivative or integrating an integral should just give us the expected derivative or appropriate integral. This way of defining a fractional derivative for the exponential function is perhaps a good introductory example, but some important questions need to be asked. Firstly, is this the only way to define the half derivative for e cx such that it has the above properties, or could we come up with a diﬀerent definition? Secondly, what happens if c < 0? For example, with c = −1, we would get that D1/2 e−x = ie−x , which is imaginary. So the fractional derivative of a real-valued function could be complex or imaginary? That sounds like dark arts to me. And finally, how does that Dν f (x) work if we are not talking about the exponential function, but if we have a polynomial or, even simpler, a constant function, like f (x) = 4? If we start with f (x) = 4 (a boring, horizontal line), we know that its first derivative is D1 f (x) = 0, so should the half derivative be something like D1/2 f (x) = 2? Then, if we half diﬀerentiate that expression again, we obtain a zero on the le-hand side (since D1 f (x) = 0) and the half derivative of a constant function (in this case g(x) = 2), on the right-hand side. But this is certainly not right! We said that the half derivative of a constant function is half the value of that constant, but now we obtain that it is zero! There is nothing worse for a mathematician than a system that is not consistent. The best way is to begin with a more formal definition. Perhaps aer having to integrate a function thousands and thousands of times, Augustin-Louis Cauchy discovered in the 19th century a way in which he could write the repeated integral of a function in a very elegant way: ∫ x 1 −n (x − t)n−1 f (t) dt. D f (x) = (n − 1)! a Not only is this a beautiful and simple formula, it also gives us a way to write any iterated integration (although to actually solve it, we would usually need to do some not-so-beautiful integration by parts). So why don’t we just change that number n to a fraction, like 9¾? Everything in that 27

spring 2016

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