论文标题
某些涉及某些指数，双曲和三角基本功能的非元素积分的分析估值以及新概率的推导概述概括γ型和正常分布
Analytical valuation of some non-elementary integrals involving some exponential, hyperbolic and trigonometric elementary functions and derivation of new probability measures generalizing the gamma-type and normal distributions
论文作者
论文摘要
涉及基本指数，双曲和三角学函数的非元素积分，$ \ int x^αe^{ηx^β} dx，\ int x^α\ cosh \ cosh \ left（ηx^ββ\ weft（ηx^β\ \ right） x^α\ cos \ left（ηx^β\右）dx $和$ \ int x^α\ sin \ sin \ left（ηx^ββ\ right）dx $，其中$α，η$和$β$是真实或复杂的常量是由汇合超细计函数$ _1f_1 $ _1 $ $ _1 $ _1 $ _1 $ _1 $ _1 $ _1 $ _1 $ _1。双曲线和EULER身份用于得出某些涉及指数，双曲线，三角函数和超几何函数$ _1F_1 $和$ _1F_2 $的身份。在评估了这些非元素积分后，还获得了一些新的概率衡量标准概括γ型和正常分布。例如，获得的广义分布可以允许进行比已经知道的统计测试更好的统计测试（例如卡方（$χ^2 $）统计测试，以及基于中央限制定理（CLT）的统计测试。
The non-elementary integrals involving elementary exponential, hyperbolic and trigonometric functions, $ \int x^αe^{ηx^β}dx, \int x^α\cosh\left(ηx^β\right)dx, \int x^α\sinh\left(ηx^β\right)dx, \int x^α\cos\left(ηx^β\right)dx$ and $\int x^α\sin\left(ηx^β\right)dx $ where $α, η$ and $β$ are real or complex constants are evaluated in terms of the confluent hypergeometric function $_1F_1$ and the hypergeometric function $_1F_2$. The hyperbolic and Euler identities are used to derive some identities involving exponential, hyperbolic, trigonometric functions and the hypergeometric functions $_1F_1$ and $_1F_2$. Having evaluated, these non-elementary integrals, some new probability measures generalizing the gamma-type and normal distributions are also obtained. The obtained generalized distributions may, for example, allow to perform better statistical tests than those already known (e.g. chi-square ($χ^2$) statistical tests and those based on central limit theorem (CLT)).
