[Merged by Bors] - feat: integration by parts on the whole real line, assuming integrability of the product #11916
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sgouezel
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| on `(a, +∞)` and continuity at `a⁺`. | ||
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| Note that such a function always has a limit at infinity, | ||
| see `tendsto_limUnder_of_hasDerivAt_of_integrableOn_Ioi`. -/ |
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I'm wondering if we should just make the RHS of this limUnder atTop f - f a, and then ensure we have simp-lemmas that can compute limUnder. Not for this PR though.
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…lity of the product (#11916) We already have that `∫ (x : ℝ), u x * v' x = b' - a' - ∫ (x : ℝ), u' x * v x` if `u * v` tends to `a'` and `b'` at minus infinity and infinity. Assuming morevoer that `u * v` is integrable, we show that it tends to `0` at minus infinity and infinity, and therefore that `∫ (x : ℝ), u x * v' x = - ∫ (x : ℝ), u' x * v x`. We also give versions with a general bilinear form instead of multiplication.
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Louddy
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…lity of the product (#11916) We already have that `∫ (x : ℝ), u x * v' x = b' - a' - ∫ (x : ℝ), u' x * v x` if `u * v` tends to `a'` and `b'` at minus infinity and infinity. Assuming morevoer that `u * v` is integrable, we show that it tends to `0` at minus infinity and infinity, and therefore that `∫ (x : ℝ), u x * v' x = - ∫ (x : ℝ), u' x * v x`. We also give versions with a general bilinear form instead of multiplication.
atarnoam
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Apr 16, 2024
…lity of the product (#11916) We already have that `∫ (x : ℝ), u x * v' x = b' - a' - ∫ (x : ℝ), u' x * v x` if `u * v` tends to `a'` and `b'` at minus infinity and infinity. Assuming morevoer that `u * v` is integrable, we show that it tends to `0` at minus infinity and infinity, and therefore that `∫ (x : ℝ), u x * v' x = - ∫ (x : ℝ), u' x * v x`. We also give versions with a general bilinear form instead of multiplication.
uniwuni
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Apr 19, 2024
…lity of the product (#11916) We already have that `∫ (x : ℝ), u x * v' x = b' - a' - ∫ (x : ℝ), u' x * v x` if `u * v` tends to `a'` and `b'` at minus infinity and infinity. Assuming morevoer that `u * v` is integrable, we show that it tends to `0` at minus infinity and infinity, and therefore that `∫ (x : ℝ), u x * v' x = - ∫ (x : ℝ), u' x * v x`. We also give versions with a general bilinear form instead of multiplication.
callesonne
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Apr 22, 2024
…lity of the product (#11916) We already have that `∫ (x : ℝ), u x * v' x = b' - a' - ∫ (x : ℝ), u' x * v x` if `u * v` tends to `a'` and `b'` at minus infinity and infinity. Assuming morevoer that `u * v` is integrable, we show that it tends to `0` at minus infinity and infinity, and therefore that `∫ (x : ℝ), u x * v' x = - ∫ (x : ℝ), u' x * v x`. We also give versions with a general bilinear form instead of multiplication.
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We already have that
∫ (x : ℝ), u x * v' x = b' - a' - ∫ (x : ℝ), u' x * v xifu * vtends toa'andb'at minus infinity and infinity. Assuming morevoer thatu * vis integrable, we show that it tends to0at minus infinity and infinity, and therefore that∫ (x : ℝ), u x * v' x = - ∫ (x : ℝ), u' x * v x. We also give versions with a general bilinear form instead of multiplication.The advantage of the version with global integrability is that it generalizes readily to higher dimensional spaces (forthcoming PR, and then applications for the Fourier transform).