廣義積分斂散性判別方法探討

廣義積分斂散性判別方法探討
摘要

                                                                                       
本文從分析被積函數(shù)本身所具的性質(zhì)出發(fā)，借助傳統(tǒng)的廣義積分斂散性判別方法，發(fā)現(xiàn)1系列更簡捷適用的新判別方法：單變量函數(shù)廣義積分方面，通過考察被積函數(shù)，結(jié)合正項級數(shù)與正函數(shù)情形下無窮區(qū)間上廣義積分的內(nèi)在聯(lián)系，給出了幾個與正項級數(shù)斂散性判別法相類似的廣義積分斂散性判別方法；多變量函數(shù)廣義積分方面，著重討論了廣義2重積分和廣義3重積分，結(jié)合被積函數(shù)的特點，運用比較判別法和柯西判別法，本文給出了判別廣義2重積分收斂的1種新方法．將單變量函數(shù)廣義積分和廣義2重積分中常用的比較判別法和柯西判別法推廣到了廣義3重積分，利用2次型理論解決了1類廣義3重積分的判斂和計算問題．
關(guān)鍵詞：廣義積分；收斂；發(fā)散

The Research of the Criterion of Convergence and Divergence of Generalized Integral
ABSTRACT

By means of traditional criterions of generalized integral’s convergence and divergence, this paper, from the analysis of integrated function’s nature, discovers a series of new criterions, which are briefer and more suitable: on the aspect of generalized integral for functions of a single variable, through the investigation of integrated function, together with the inner relationship between positive series and generalized integral in infinite interval under the condition of positive function, it gives several criterions of generalized integral’s convergence and divergence, which are similar to positive series’ criterions of convergence and divergence. On the aspect of generalized integral for functions of several variables, it emphasizes on generalized integral for functions of two variables and generalized integral for functions of three variables. By using the comparison test and the Cauchy test, along with the characters of integrated function, this paper gives a new way to determine the convergence of generalized integral for functions of two variables as well as a new way to determine its divergence. The comparison test and the Cauchy test which are often used in generalized integral for functions of a single variable and that of two variables are generalized to generalized integral for functions of three variables, and it uses the quadratic form theory to determine and calculate generalized integral for functions of three variables.
Key words: generalized integral; convergence; divergence

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