Finite difference

Definition

Let $f \colon \R \to \R$ be a function, and let $a, b \in \R$ with $a \ne b$. The finite difference of $f$ with offsets $a$ and $b$ is the function $$x \mapsto f(x + b) - f(x + a).$$ For a step $h > 0$, the three standard cases are:

1. the forward difference $\Delta_h f(x) = f(x + h) - f(x)$, with $a = 0$ and $b = h$;
2. the backward difference $\nabla_h f(x) = f(x) - f(x - h)$, with $a = -h$ and $b = 0$;
3. the central difference $\delta_h f(x) = f(x + \tfrac{h}{2}) - f(x - \tfrac{h}{2})$, with $a = -\tfrac{h}{2}$ and $b = \tfrac{h}{2}$.