论文标题
评估一些涉及广义超测量函数的非元素积分
Evaluation of some non-elementary integrals involving the generalized hypergeometric function with some applications
论文作者
论文摘要
无限的积分$$ \ int x^αe^{ηx^β} \，_ pf_q（a_1，a_2，a_2，\ cdot \ cdot \ cdot \ cdot \ cdot \ cdot a_p; b_1，b_2，b_2，\ cdot \ cdot \ cdot \ cdot \ cdot \ cdot \ cdot \ cdot，b_q; b_q; b_q;λx^upx^um $ γ\ ne0 $是真实的或复杂的常数，$ _PF_Q $是广义的超几何函数，根据涉及广义超几何函数的无限序列评估。指数函数$ e^{ηx^β} $的相关积分要么被双曲函数$ \ cosh \ left（ηx^β\ right）$或$ \ sinh \ left（ηx^ββ\ right）$，或sinosoidal函数$ \ cos \ cos \ weft（ $ \ sin \ left（ηx^β\ right）$也通过无限序列进行评估，涉及广义超几何函数$ _PF_Q $。对应用分析进行了一些应用示例，其中评估了一些新的傅立叶和拉普拉斯积分（或变换）。在短波限制中，ORR-Sommerfeld方程（具有线性流动背景）的分析解决方案用某些无限序列表示涉及超几何序列$ _2F_3 $。利用双曲线和欧拉的身份，还得出了一些有趣的串联身份，涉及指数，双曲线，三角函数和广义超几何函数。
The indefinite integral $$ \int x^αe^{ηx^β}\,_pF_q (a_1, a_2, \cdot\cdot\cdot a_p; b_1, b_2, \cdot\cdot\cdot, b_q; λx^γ)dx, $$ where $α, η, β, λ, γ\ne0$ are real or complex constants and $_pF_q$ is the generalized hypergeometric function, is evaluated in terms of an infinite series involving the generalized hypergeometric function. Related integrals in which the exponential function $e^{ηx^β}$ is either replaced by the hyperbolic function $\cosh\left(ηx^β\right)$ or $\sinh\left(ηx^β\right)$, or the sinusoidal function $\cos\left(ηx^β\right)$ or $\sin\left(ηx^β\right)$, are also evaluated in terms of infinite series involving the generalized hypergeometric function $_pF_q$. Some application examples from applied analysis, in which some new Fourier and Laplace integrals (or transforms) are evaluated, are given. The analytical solution of the Orr-Sommerfeld equation (with a linear mean flow background) in the short-wave limit is expressed in terms of some infinite series involving the hypergeometric series $_2F_3$. Making use of the hyperbolic and Euler identities, some interesting series identities involving exponential, hyperbolic, trigonometric functions and the generalized hypergeometric function are also derived.
