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function_integral_on_ℝⁿ [2016/02/06 15:07] nikolaj |
function_integral_on_ℝⁿ [2017/01/05 00:00] nikolaj |
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See references. | See references. | ||
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+ | == Kernel of he integral == | ||
+ | A linear combination of functions that are zero under an integral are again zero. | ||
+ | |||
+ | Special case | ||
+ | |||
+ | $$\int_{-a}^a E(x) \left( \dfrac{1}{2} + \sum_{k=0}^\infty c_k U_k(x)^{2k+1} \right) \,{\mathrm d}x = \int_0^a E(x) \,{\mathrm d}x$$ | ||
+ | |||
+ | e.g. all $U_k$ the same and $c_k$ so that you get $\frac{1}{1\pm e^{y}}$: | ||
+ | |||
+ | $$\int_{-a}^a E(x) \dfrac{1}{1\pm {\mathrm e}^{U(x)}}\,{\mathrm d}x = \int_0^a E(x) \,{\mathrm d}x$$ | ||
+ | |||
+ | $$\int_{-a}^a f(x^2) \dfrac{1}{1 + {\mathrm e}^{x^2\sin(x)}}\,{\mathrm d}x = \int_0^a f(x^2) \,{\mathrm d}x$$ | ||
=== References === | === References === |