What level of math is needed to learn fractional calculus?

Question

I was skimming through wikipedia pages and stumbled upon the fractional calculus page. My interest increased when I noticed it has applications in physics. I was wondering as an undergraduate who's highest level of math is introductory real analysis, what would I need to understand this topic? Are there any textbooks that can be recommended?

Answer 1

You may be able to see some definitions of fractional derivatives and integrals with just an introductory real analysis background. But the general theory of fractional derivatives is a functional analysis topic.

To get a brief glimpse into fractional integration/differentiation you can just take the various definitions provided in the Wikipedia article (i.e. Riemann-Liouville, Caputo) as a starting point. These definitions should quite accessible (if rather unmotivated) with an introductory real analysis background.

The general theory for fractional derivatives falls into something we call pseudodifferential operators and requires more mathematical sophistication. At the very least, you need to be familiar with the Fourier transform and function spaces. Pseudodifferential operators can be developed either in the more restricted context of Sobolev spaces (where you most clearly see the application to PDEs), or in the slightly more general context of distribution theory (and beyond, getting into microlocal analysis, where you start to see its enormous power for linear PDEs). Both are topics that can come in a functional analysis course after a second course in real analysis (i.e. measure theory and integration).

Answer 2

Fractional calculus is as accessible as integer calculus. Dressing it up with twenty-five-dollar words to describe 25 cent concepts does little to improve understanding or encourage learning. The fraction calculus by Oldham and Spanier offers a broad introduction to the subject. It contains derivations, historical background, numerical techniques and applications. It is sold by Dover and Amazon at a reasonable price.