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function_integral_on_ℝⁿ [2016/02/06 15:07] nikolaj |
function_integral_on_ℝⁿ [2016/03/28 20:29] 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 | ||
| + | |||
| + | $$\int_{-a}^a E(x) \left( 1 + \sum_{k=0}^\infty c_k U_k(x)^{2k+1} \right) = \int_0^a E(x) \,{\mathrm d}x$$ | ||
| + | |||
| + | e.g. all $U_k$ the same and $c_k$ so that you get $\frac{1}{1\pm e^{y}}$: | ||
| + | |||
| + | $$\int_{-a}^a E(x) \dfrac{1}{1\pm {\mathrm e}^{U(x)}}\,{\mathrm d}x = \int_0^a E(x) \,{\mathrm d}x$$ | ||
| + | |||
| + | $$\int_{-a}^a f(x^2) \dfrac{1}{1 + {\mathrm e}^{x^2\sin(x)}}\,{\mathrm d}x = \int_0^a f(x^2) \,{\mathrm d}x$$ | ||
| === References === | === References === | ||
