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:
Definition: Difference Operator
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 $$\Delta f(x)=f(x-1)-f(x).$$
Notes
- $D$ 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 $$\frac {f(x+h)-f(x)}{h}=g(y+1)-g(y)=\Delta g(y).$$
Mentioned in:
Definitions: 1
Examples: 2
Proofs: 3 4 5 6 7 8 9
Propositions: 10 11 12
Theorems: 13
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References
Bibliography
- Graham L. Ronald, Knuth E. Donald, Patashnik Oren: "Concrete Mathematics", Addison-Wesley, 1994, 2nd Edition
- Miller, Kenneth S.: "An Introduction to the Calculus of Finite Differences And Difference Equations", Dover Publications, Inc, 1960
- Bool, George: "A Treatise on the Calculus of Finite Differences", Dover Publications, Inc., 1960
