一类高振荡超奇异积分的分析与计算  

Analysis and Calculation of a Class of Hypersingular Integrals with High Oscillation

作　　者：王箬夏 康洪朝 WANG Ruoxia;KANG Hongchao(School of Sciences,Hangzhou Dianzi University,Hangzhou Zhejiang 310018,China)

机构地区：[1]杭州电子科技大学理学院,浙江杭州310018

出　　处：《杭州电子科技大学学报（自然科学版）》2023年第4期66-71,共6页Journal of Hangzhou Dianzi University：Natural Sciences

基　　金：国家自然科学基金资助项目(11301125,11971138);浙江省自然科学基金资助项目(LY22A010002,LY18A010009)。

摘　　要：运用复积分方法计算一类高振荡超奇异积分∫b a f(x)/(x-σ)v+1 e iωx d x,其中a<σ<b,ω是较大的实数,v是正整数,i是虚数单位,f(x)在包含[a,b]的足够大的复区域内是解析的。首先,将∫b a f(x)(x-σ)v+1 e iωx d x看作高振荡柯西主值奇异积分∫b a f(x)/x-σe iωx d x的v阶导数形式;然后,根据解析延拓,将其转化为2个无穷积分,因获得的无穷积分的被积函数是非振荡且指数快速衰减的,故使用高斯拉盖尔积分法则进行高效计算;最后,针对ω负次幂的误差展开分析,并通过数值实验验证了复积分方法的高效性和精确性。Using the complex integration method to calculate a class of highly oscillatory hypersingular integral∫b a f(x)/(x-σ)v+1 e iωx d x,where a<σ<b,ωis a large real number,v is a positive integer,i is an imaginary unit,and is analytic in a sufficiently large complex region containing[a,b].Firstly,it is regarded as the v-order derivative form of highly oscillatory Cauchy principal value integral∫b a f(x)/x-σe iωx d x.Then,it is transformed into 2 infinite integrals according to analytic continuation,since the integrands of the obtained infinite integrals are non-oscillatory and exponentially decay rapidly,which can be efficiently calculated by using the Gaussian-Laguerre integral rule.Finally,not only the error analysis in inverse powers ofωis given,but al2so the numerical experiments can verify the efficiency and accuracy of the complex integration method.

关 键 词：高振荡积分 超奇异 复积分法 高斯拉盖尔积分法则 

分 类 号：O241.4[理学—计算数学]