双曲函数表示级数封闭形和式  

Hyperbolic Function Representations of Closed Form of Class Series

作　　者：陈艳丽[1] 张来萍[1] 及万会[1] CHEN Yan-li;ZHANG Lai-ping;JI Wan-hui(Department of Basic Courses,Yinchuan Energy College,Yongning,Yinchuan 750105,China)

机构地区：[1]银川能源学院基础部,银川永宁75010

出　　处：《河北北方学院学报（自然科学版）》2018年第9期1-6,16,共7页Journal of Hebei North University:Natural Science Edition

摘　　要：国内外数论方面有关级数和式问题的研究有大量不同的方法和结果,针对如下级数和式问题:设c∈Q\{0},∑n∈z(1/n^2+c^2);∑n∈z1/((n^2+c^2)~2);∑n∈z1/(n^2-c^2);∑n∈z1/(n^4+c^4);∑n∈z1/(n^4-c^4);∞∑-∞((-1)~n)/(n^2+c^2);∞∑n=1((-1)~n)(n^3sinh(nπ)),提出应用双曲函数研究数论的性质,给出应用双曲函数表示的级数封闭形和式。利用复数棣莫弗公式、双曲函数特性、傅里叶变换以及复变函数的方法,给出这类级数和式的双曲函数表达形式。并应用留数定理对所给出的这类应用双曲函数表示的级数封闭形和式作出证明。There are a lot of different methods and results in the study of series and formula problems in the theory of numbers at home and abroad.For the following sums of series:∑n∈z1/n^2+c^2;∑n∈z1/n^2-c^2;∞∑-∞(-1)^n/n^2+c^2;∞∑n=1(-1)n^/n^3sinh(nπ),the properties of number theory are studied by using hyperbolic functions,and the closed form sum of series expressed by hyperbolic functions is given.The hyperbolic function expression of this kind of series sum is given by using the complex Dimmer formula,the characteristic of hyperbolic function,Fourier transform and the method of complex variable function.The closed form sum expressed by hyperbolic function is given,and the closed form expressed by hyperbolic function is proved by residue theorem.

关 键 词：傅里叶变换 复变函数 双曲函数 封闭形 级数 

分 类 号：O173[理学—数学]