---
tag: T73XK15SA2
kind: concept
name: Fractional calculus
msc: [26A33]
sources:
  - url: https://en.wikipedia.org/wiki/Fractional_calculus
same_as:
  wikidata: Q1339058
  wikipedia: Fractional calculus
---

## Summary

Fractional calculus studies integrals and [[derivatives|0V1EJCT8G6]] of non-integer order, such as a "half derivative".
Its basic operator is the [[Riemann–Liouville integral|JTKCZ44NN3]], which extends repeated integration to any real order $\alpha > 0$.
Fractional derivatives, such as the Riemann–Liouville, [[Caputo|6VMGQYF8RW]] and [[Grünwald–Letnikov|M9Z8VM4K1Z]] derivatives, are built from it or from limits of differences.
These operators model memory effects and anomalous diffusion, and appear in integral equations such as Abel's equation.

## Explanations

### L1 human

Differentiating once, twice or three times is familiar; fractional calculus asks what it means to differentiate half a time.
The idea is to find a formula for the $n$-th integral that makes sense when $n$ is not a whole number.
[[Cauchy's formula for repeated integration|EYXG11V76E]] gives such a formula, with the factorial replaced by the [[gamma function|0H5SYBSED9]].
For example, the half integral of $1$ from $0$ is $2\sqrt{x/\pi}$, and doing the half integral twice gives the ordinary integral $x$.