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  "body": "\n## Definition\n\nLet $f \\colon \\R \\to \\R$ be a function and let $h \u003e 0$.\nThe *central difference* of $f$ with step $h$ is the function $\\delta_h f \\colon \\R \\to \\R$ given by\n$$\\delta_h f(x) = f\\left(x + \\tfrac{h}{2}\\right) - f\\left(x - \\tfrac{h}{2}\\right).$$\n\n## Remarks\n\nIt is the [[finite difference|7JSQZ126BP]] with offsets $-\\tfrac{h}{2}$ and $\\tfrac{h}{2}$.\nExample: for $f(x) = x^3$, $\\delta_h f(x) = 3hx^2 + \\tfrac{h^3}{4}$, by expanding both cubes.\nNon-example: $f(x + h) - f(x)$ is not centred at $x$, so it is the forward difference and not the central difference.\nFor sequences one often uses the version $f(x + h) - f(x - h)$ with step $2h$, which avoids half-integer points.\nIf $f$ is three times continuously differentiable, Taylor's theorem gives $\\frac{f(x + h) - f(x - h)}{2h} = f'(x) + O(h^2)$ as $h \\to 0$.\n",
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