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| Both sides previous revision Previous revision Next revision | Previous revision | ||
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function_integral_on_ℝⁿ [2016/03/28 20:22] nikolaj |
function_integral_on_ℝⁿ [2017/01/05 00:00] nikolaj |
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| Special case | Special case | ||
| - | $$\int_{-a}^a E(x) \left( 1 + \sum_{k=0}^\infty c_k U_k(x)^{2k+1} \right) = \int_0^a E(x) \,{\mathrm d}x$$ | + | $$\int_{-a}^a E(x) \left( \dfrac{1}{2} + \sum_{k=0}^\infty c_k U_k(x)^{2k+1} \right) \,{\mathrm d}x = \int_0^a E(x) \,{\mathrm d}x$$ |
| e.g. all $U_k$ the same and $c_k$ so that you get $\frac{1}{1\pm e^{y}}$: | e.g. all $U_k$ the same and $c_k$ so that you get $\frac{1}{1\pm e^{y}}$: | ||
| $$\int_{-a}^a E(x) \dfrac{1}{1\pm {\mathrm e}^{U(x)}}\,{\mathrm d}x = \int_0^a E(x) \,{\mathrm d}x$$ | $$\int_{-a}^a E(x) \dfrac{1}{1\pm {\mathrm e}^{U(x)}}\,{\mathrm d}x = \int_0^a E(x) \,{\mathrm d}x$$ | ||
| + | |||
| + | $$\int_{-a}^a f(x^2) \dfrac{1}{1 + {\mathrm e}^{x^2\sin(x)}}\,{\mathrm d}x = \int_0^a f(x^2) \,{\mathrm d}x$$ | ||
| === References === | === References === | ||
