# Discrete Calculus

1. Nov 30, 2004

### Eppur si muove

Is there such thing as discrete calculus? Or are there general rules to find derivatives and integrals of functions whose domains are restricted to integers or some other discrete values?

2. Nov 30, 2004

### Hurkyl

Staff Emeritus
An integral over a discrete set is simply a sum! In the general case, the integral of a function $f$ with respect to a measure $\mu$ can be computed by:

$$\int_A f d\mu = \sum_{a \in A} f(a) \mu(a)$$

There is a discrete analog of a derivative called a difference:

$$\Delta_hf(x) = f(x + h) - f(x)$$

(when h is omitted, assume it's 1)

And difference equations have many similarities with differential equations. For example, one can "solve" for the Fibonacci sequence which is defined by a linear second-order homogenous difference equation:

$$\Delta^2 F + \Delta F - F = 0 | F(0) = 0, F(1) = 1$$

whose solution technique is directly analogous to that of similar differential equations: (use $F(r) = a^r$ as a putative solution, get two linearly independent solutions, and take a linear combination that satisfies the initial conditions)

There's a more general concept here called a skew derivation (or $\sigma$-derivation) of which both the ordinary derivative and this finite difference are examples.

And, of course, there's the antidifference operator, also called the summation operator, which bears a similar to indefinite integrals. For instance, you can even do summation by parts.

Last edited: Nov 30, 2004
3. Dec 1, 2004

### HallsofIvy

It's more often called "finite differences" rather than "discrete calculus".

Try a google search on "finite differences". Boole wrote a book on it that is still published by Dover.