Half Derivatives: An Operator Theory Prospective

Half Derivative Definition

When we’re taught calculus, we’re taught second derivatives are a derivative of a derivative, and the third derivative is the derivative of that and so on.

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  • The exponential
  • Sine and cosine
  • Polynomials

Calculating the Half Derivative: Special Functions

In general half derivatives can be difficult to calculate, but let’s look at one case where it’s not difficult: The exponential function is an “eigenfunction” of the derivative, meaning that its derivative is the same function up to a constant multiplier.

Matrix Operators

Before we go forward, let’s first talk about matrices. The reason is that we’ve been talking about derivatives as linear operators, which makes them generalizations of matrices. (We’ve seen this analogy already when we mentioned eigenfunctions.) Basic operator theory is motivated from things known about matrices. For example, the half derivative is the square root of the derivative in the matrix sense: If the square root of a matrix M is A then A(Av) = Mv for any vector v. Similarly if we call the square root of the derivative “H”, then H(Hf) = Df; so H is the half-derivative.

Singular Value Decomposition

If you knew already that Taylor series apply to matrices, then this is a natural first approach to taking a square root. But Taylor series are difficult to caclulate; in practice you could only ever get an approximation, because you can’t calculate all the terms. There is however a more clever way to take the square root, which we will explore in this section.

Fourier Transform

Returning now to the derivative. Derivatives are linear operators on the vector space of smooth functions. But unlike 2-D vectors like we used in our example, function spaces are infinite dimensional, meaning that they don’t have any finite basis.

Fractional Derivatives: Polynomials

We’re now finally ready to calculate the fractional derivative of the power function, x^m. We want to do this calculation for a general m, so we will only consider positive x, because x^m is not defined when x is negative and m is irrational.

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