Difference Calculus
Most people have heard of Differential Calculus, but few have heard of Difference Calculus. In the rare times it does get mentioned it's described as an "approximation" to Differential Calculus. That's a pity, because Difference Calculus deserves better!
So what is it?
It's easily explained if you remember the definition of a derivative in Differential Calculus. For a function f(x) the derivative df(x)/dx is defined as..
df(x)/dx=Limit as c approaches 0 of (f(x+c)-f(x))/c
It turns out that for this limit to exist f(x) has to be a very "well behaved" function. Not all functions are, so df(x)/dx does not exist for many functions. Which means that Differential Calculus cannot be used with these functions.
Difference Calculus gets around this problem in an incredibly simple way, it just removes the limit from the definition of df(x)/dx to define the "difference" Df(x) as..
Df(x)=(f(x+c)-f(x))/c
But this definition is not as simple as it looks. Notice that in the definition of df(x)/dx the c never appears in the result, it's simply used to define the limit. This is not true for Df(x), where c actually appears in the result, and so different choices of c will give different values for Df(x). This is not a bad thing, but it's something that should be noted. To be strictly accurate we should denote the difference as Dcf(x) because it depends on c, but it's usually just denoted by Df(x) and some value of c is assumed.
Of course, for the case of c=1 we get the simple expression..
Df(x)=f(x+1)-f(x)
Let's try an example..
The function f(x)=e^x is famous in Differential Calculus because it's equal to its own derivative, df(x)/dx=f(x)
This is not true in Difference Calculus, because the difference is..
Df(x)=f(x)*(e^c-1)
As we noted above, the result depends on c. If we want to get the same result as differential calculus, Df(x)=f(x), we have to use a specific value of c such that..
(e^c-1)=1 which means c=ln(2)=0.69314718056..
This is a hint that Difference Calculus has its own properties and is more than just an "approximation" to Differential Calculus.
For functions of an integer variable, f(n) where n=1,2,3,4,.. the value c=1 is natural and the difference is Df(n)=f(n+1)-f(n)
Equations involving Df(x) are called difference equations and are the equivalent of differential equations in Differential Calculus. Here's a simple example..
Df(x)=-2*f(x) for c=1, which is the simple difference equation f(x+1)+f(x)=0
One solution to this difference equation is f(x)=a*cos(pi*x), where a is an arbitrary constant. So even a very simple difference equation has a "wave-like" solution! This is another hint that Difference Calculus has its own properties distinct from Differential Calculus.
In Differential Calculus we can define higher order derivatives. Can we do this in Difference Calculus? Yes! For example, the second order difference is just the difference of the first order difference..
D(Df(x))=(f(x+2*c)-2*f(x+c)+f(x))/(c^2) and for the case of c=1 this is the simple expression f(x+2)-2*f(x+1)+f(x)
Higher order differences simply use the value of the function at more points. The first order difference uses x and x+1. The second order differences uses x, x+1 and x+2. This continues for higher order differences.
Difference Calculus has very broad applicability. In fact, I like to think of Differential Calculus as just a special case of Difference Calculus!
More to explore: "The Theory of Finite Differences" by C. Jordan, first published in Budapest in 1938. This book give a complete mathematical treatment of Difference Calculus.
Content written and posted by Ken Abbott abbottsystems@gmail.com