“reduction formula”的概念、定义、翻译、参考文献-科学参考

reduction formula

Mathematics

单词	reduction formula
释义	reduction formula Mathematics Let I_n be some quantity that is dependent upon the natural number n. It may be possible to establish some general formula, expressing I_n in terms of some of the quantities I_n −1, I_n −2,…. Such a formula is a reduction formula and can be used to evaluate I_n for a particular value of n. The method is particularly useful in integration. For example, if $I_{n} = \int_{0}^{π / 2} {sin}^{n} x d x,$ it can be shown, by integration by parts, that $I_{n} = (\frac{n - 1}{n}) I_{n - 2} for n \geq 2.$ As I₀ = π‎/2, the reduction formula can be used, for example, to find that $I_{6} = \frac{5}{6} I_{4} = \frac{5}{6} \times \frac{3}{4} I_{2} = \frac{5}{6} \times \frac{3}{4} \times \frac{1}{2} I_{0} = \frac{5}{6} \times \frac{3}{4} \times \frac{1}{2} \times \frac{π}{2} = \frac{5 π}{32} .$
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