Discrete Calculus: Derivatives & Integrals

[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?

 

[Hurkyl]

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.

 

[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.