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function_integral_on_ℝⁿ [2016/02/06 15:06] nikolaj |
function_integral_on_ℝⁿ [2016/03/28 20:22] nikolaj |
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^ $ {\mathrm d}\left(\int_{v(y)}^{w(y)}\,f(x)\,{\mathrm d}x\right) = f(v(y))\,{\mathrm d}v(y)-f(w(y))\,{\mathrm d}w(y) $ ^ | ^ $ {\mathrm d}\left(\int_{v(y)}^{w(y)}\,f(x)\,{\mathrm d}x\right) = f(v(y))\,{\mathrm d}v(y)-f(w(y))\,{\mathrm d}w(y) $ ^ | ||
- | For $f$ convex and $\langle f\rangle_{[a,b]}:=\dfrac{1}{b - a}\int_a^b f(x)\,{\mathrm d}x$ | + | For $f$ convex and |
+ | |||
+ | $\langle f\rangle_{[a,b]}:=\dfrac{1}{b - a}\int_a^b f(x)\,{\mathrm d}x$ | ||
^ $\dfrac{f(a) + f(b)}{2} \ge \langle f\rangle_{[a,b]} \ge f \left(\dfrac{a+b}{2}\right) $ ^ | ^ $\dfrac{f(a) + f(b)}{2} \ge \langle f\rangle_{[a,b]} \ge f \left(\dfrac{a+b}{2}\right) $ ^ | ||
- | ==== Parents ==== | + | See references. |
+ | |||
+ | == Kernel of he integral == | ||
+ | A linear combination of functions that are zero under an integral are again zero. | ||
+ | |||
+ | Special case | ||
+ | |||
+ | $$\int_{-a}^a E(x) \left( 1 + \sum_{k=0}^\infty c_k U_k(x)^{2k+1} \right) = \int_0^a E(x) \,{\mathrm d}x$$ | ||
+ | |||
+ | e.g. all $U_k$ the same and $c_k$ so that you get $\frac{1}{1\pm e^{y}}$: | ||
+ | |||
+ | $$\int_{-a}^a E(x) \dfrac{1}{1\pm {\mathrm e}^{U(x)}}\,{\mathrm d}x = \int_0^a E(x) \,{\mathrm d}x$$ | ||
+ | |||
+ | === References === | ||
+ | Wikipedia: | ||
+ | [[https://en.wikipedia.org/wiki/Hermite%E2%80%93Hadamard_inequality|Hermite–Hadamard inequality]] | ||
+ | |||
+ | ----- | ||
=== Subset of === | === Subset of === | ||
[[Function integral]] | [[Function integral]] |