Calculus is the manipulation of one basic operator: the derivative or . This operator operates on functions and by repeatedly applying it, you can get higher order derivatives. It’s inverse operator is known as the integral. Similar to matrix operators which have eigenvalues and eigenvectors, this operator also has eigenvalues and eigenfunctions. The eigenfunction is the function which only goes through some scalar change when acted on by the operator. This scalar that the function is scaled by is called the eigenvalue of the eigenfunction. For the derivative operator, the eigenfunctions would be any function of the form because . The eigenvalue would be . This means applying the derivative operator to this function times only results in multiplying the original function by its eigenvalue times. It is explicitly written below.
This means the half derivative of the exponential function, in some sense, is . Using this, one can even find fractional derivatives of trigonometric functions by writing them as complex exponentials. The half derivative of sine, for example, is shown below.
In fact, a general formula for both the derivatives of sine and cosine are given below using eigenfunctions.
This seems to only work for trigonometric and exponential functions. However, using a Fourier transform, one can rewrite any function in terms of exponentials. The Fourier transform of any function is shown below.
A general method can then be created for the derivative of a function as shown below.
This can also be done with Laplace transforms which would give the relation .
Through a similar process, one can find out how to calculate fractional Fourier transforms. The eigenfunction of the fractional Fourier transforms is actually the Gaussian so . One can put any function in terms of Gaussians using Hermite transforms.
There are actually much better ways of calculating fractional derivatives through fractional integrals and even fractional Fourier transforms which are used for more rigorous calculations in the field of fractional calculus but these are just some neat methods to do so.
If you want to learn more or see where I learned this from, watch Ahmed Isam.