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  "body": "\n## Definition\n\nLet $f \\colon \\R \\to \\R$ be a function, and let $\\Delta$ be the [[forward difference operator|TXTR68V4R1]] with step $1$.\nAn *indefinite sum* of $f$ is a function $F \\colon \\R \\to \\R$ with $\\Delta F = f$, that is,\n$$F(x + 1) - F(x) = f(x) \\quad \\text{for all } x \\in \\R.$$\nAnother name is *antidifference*.\nThe same definition applies to functions $f \\colon \\Z \\to \\R$, with $F \\colon \\Z \\to \\R$.\n\n## Remarks\n\nCommon notations are $\\Delta^{-1} f$ and $\\sum_x f(x)$.\nExample: $F(x) = 2^x$ is an indefinite sum of $f(x) = 2^x$, since $2^{x+1} - 2^x = 2^x$.\nNon-example: $F(x) = x^2/2$ is not an indefinite sum of $f(x) = x$, since $F(x + 1) - F(x) = x + \\tfrac{1}{2}$.\nIndefinite sums are not unique.\nOn $\\Z$, two indefinite sums of $f$ differ by a constant.\nOn $\\R$, two indefinite sums of $f$ differ by a function $C$ with $C(x + 1) = C(x)$ for all $x$, such as $\\sin(2\\pi x)$.\n",
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