## Generalisations of Calculus

There are various ways in which the integral and differential operators may be generalised or varied.

The basic notions and notations are

$D_x(f)(x) = \dfrac{d}{dx}f(x)$;

$I_x(f)(x) + K = \int f(x)\,dx$; and

$I_{x;a,b}(f)(x) = \int_{a}^{b} f(x)\,dx$.

### Fractional Calculus

The notation

$D_x^n(f)(x) = \dfrac{d^n}{dx^n}f(x)$

exists for repeated differentiation. In the standard calculus, $n \in \mathbb{N}$, with $D^n = I^{-n}$ for $n < 0$.

This notion may be generalised for $n \in \mathbb{Q}$ or $n \in \mathbb{R}$. So, $D_x^{1/2}$ would be an operator that differentiates a function half a time with respect to $x$. This operator would be such that:

$D_x^{1/2}(D_x^{1/2}(f)) = D_x(f)$.

There are many papers, books and websites on the subject. (See for example Wikipedia.)

It seems that this can even be take further so that $n \in \mathbb{N}[i]$ or even $n \in \mathbb{C} = \mathbb{R}[i]$.

### Generalised Calculus

There is a relationship between the discrete and continuous operators; for example, $\sum_{k=a}^{b}f(x)$ is the discrete sum, and $\int_{a}^{b}f(x)\,dx$ is the continuous sum.

Generalised calculus looks for such correspondences with other operators, and in particular, the discrete product $\prod_{k=a}^{b}f(x)$.

(See Math2.org.)

### Finite Calculus or Umbral Calculus

Ordinary calculus is sometimes known as infinitesimal calculus. This may be contrasted with the finite calculus.

Here, the finite operators

$\Delta f(x)$ corresponding to $Df(x)$

$\sum_a^b f(x)\,\delta x$ corresponding to $\int_a^b f(x)\,dx$

are developed.

### Others?

There are algebraic or topological ways to generalise the calculus. For example, you might consider operating over other differential fields rather the field of complex numbers.

However, I’m primarily interested in generalisations of the infinitesimal calculus over the complex numbers (or common subsets, such as real numbers or integers).