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Fractional calculus The calculus of witchcraft and wizardry Yusuke Kawasaki, licensed under Creative Commons CC BY 2.0

Rafael Prieto Curiel


 a function is usually regarded as a discrete operation: we use the first derivative of a function to determine the slope of the line that is tangent to it, and we differentiate twice if we want to know the curvature. We can even differentiate a function negative times—ie integrate it—and thanks to that we measure the area under a curve. But why stop there? Is calculus limited to discrete operations, or is there a way to define the half derivative of a function? Is there even an interpretation or an application of the half derivative? Fractional calculus is a concept as old as the traditional version of calculus, but if we have always thought about things using only whole numbers then suddenly using fractions might seem like taking the Hogwarts Express from King’s Cross station. However, fractional calculus opens up a whole new area of beautiful and magical maths. How do we interpret the half derivative of a function? Since we are only halfway between the first derivative of a function and not differentiating it at all, then maybe the result should also be somewhere between the two? For example, if we have the function ecx , with c > 0, then we can write any derivative of the function as Dn e cx = c n e cx , which works for n = 1, 2, . . . , but also works for integrating the function (n = −1) and doing nothing to it at all (n = 0). As an aside, the symbol Dn might seem like a weird notation to represent the nth derivative, or D−n to represent the nth integral of a function, but it’s just an easy


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