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  "body": "\n## Definition\n\nLet $C^1[0, \\infty)$ be the space of continuously differentiable functions $f \\colon [0, \\infty) \\to \\C$, with the one-sided [[derivative|0V1EJCT8G6]] at $0$.\nThe *differentiation operator* $p \\colon C^1[0, \\infty) \\to C[0, \\infty)$ is $p f = f'$.\nIts companion, the integration operator $p^{-1} \\colon C[0, \\infty) \\to C^1[0, \\infty)$, is\n$$(p^{-1} f)(t) = \\int_0^t f(\\tau) \\, d\\tau.$$\n\n## Remarks\n\nThe two operators satisfy\n$$p \\, p^{-1} f = f \\quad (f \\in C[0, \\infty)), \\qquad p^{-1} p f = f - f(0) \\quad (f \\in C^1[0, \\infty)).$$\nThe first identity is the fundamental theorem of calculus: $p^{-1} f$ is the [[antiderivative|0BYWRWFT4F]] of $f$ that vanishes at $0$.\nSo $p^{-1}$ is a right inverse of $p$ but not a left inverse.\n\nExample: for $f(t) = e^{at}$, $p f = a f$, and $p^{-1} f = (e^{at} - 1)/a$ for $a \\ne 0$; then $p \\, p^{-1} f = e^{at} = f$.\n\nNon-example of a two-sided inverse: for the constant $f = 1$, $p^{-1} p f = p^{-1} 0 = 0 \\ne 1$; the clause $p^{-1} p = \\id$ fails exactly because $f(0) \\ne 0$.\n\nIn the Mikusiński calculus, the integration operator is convolution with the constant $1$, and the corresponding element $s = 1/l$ of the field of operators satisfies $s f = f' + f(0)$, which repairs the defect above.\n",
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