In analogy to the derivative $f’(x)=\lim_{h\to 0}\frac {f(x+h)-f(x)}{h},$ we introduce the *difference operator* as follows:

Let \(D\subseteq\mathbb R\) (\(D\) is a subset of real numbers) and let \(f:D\to\mathbb R\) be a function. The **difference operator** $\Delta f$ is defined by $$\Delta f(x)=f(x-1)-f(x).$$

- $D$ is assumed to contain both, $x$ and $x+1.$
- Unlike the derivative $f’(x)$, the difference operator $\Delta f(x)$ always exists, provided, $f(x)$ is defined.
- Occasionally, we can reduce a given differential operator $\frac {f(x+h)-f(x)}{h}$ by replacing $x$ by the
*normalized variable*$y:=\frac xh$ and $f(x)$ by the normalized function $g(x/h):= f(x)/h.$ Then $$\frac {f(x+h)-f(x)}{h}=g(y+1)-g(y)=\Delta g(y).$$

| | | | | created: 2020-03-23 20:57:53 | modified: 2020-03-23 21:13:45 | by: *bookofproofs* | references: [1112], [8404], [8405]

[8405] **Bool, George**: “A Treatise on the Calculus of Finite Differences”, Dover Publications, Inc., 0

[1112] **Graham L. Ronald, Knuth E. Donald, Patashnik Oren**: “Concrete Mathematics”, Addison-Wesley, 1994, 2nd Edition

[8404] **Miller, Kenneth S.**: “An Introduction to the Calculus of Finite Differences And Difference Equations”, Dover Publications, Inc, 0