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  "body": "\n## Definition\n\nLet $f \\colon \\R \\to \\R$ be a [[function|0XHDTE5YMZ]], and let $a, b \\in \\R$ with $a \\ne b$.\nThe *finite difference* of $f$ with offsets $a$ and $b$ is the function\n$$x \\mapsto f(x + b) - f(x + a).$$\nFor a step $h \u003e 0$, the three standard cases are:\n\n1. the forward difference $\\Delta_h f(x) = f(x + h) - f(x)$, with $a = 0$ and $b = h$;\n2. the backward difference $\\nabla_h f(x) = f(x) - f(x - h)$, with $a = -h$ and $b = 0$;\n3. the central difference $\\delta_h f(x) = f(x + \\tfrac{h}{2}) - f(x - \\tfrac{h}{2})$, with $a = -\\tfrac{h}{2}$ and $b = \\tfrac{h}{2}$.\n\n## Remarks\n\nExample: for $f(x) = x^3$ and $h = 1$, the forward difference is $(x+1)^3 - x^3 = 3x^2 + 3x + 1$.\nNon-example: $f(x + h) + f(x)$ is not a finite difference, since it is a sum of values and not a difference.\nDividing by $b - a$ gives the difference quotient $\\frac{f(x + b) - f(x + a)}{b - a}$.\nThe same formulas make sense for functions from an abelian group to an abelian group, and for sequences, where the step is $1$.\nSee the [[forward difference operator|8E0EBA8X07]], the [[backward difference operator|MK5JGVDPS9]] and the [[central difference|BVGYDP6YKF]].\n",
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