integral#Integration by parts#The indefinite integral#Substitution#Absolute integrability#Convergence theorems#Integration over elementary sets#Integrals of vector, matrix and complex functions#Relation to the
To sum up, in this blog, we first define and explore the Riemann-Stieltjes integral, then uses this integration techniques to solve problems via asymptotic expansion. Lastly, we provide a proof for the prime number theorem. 编辑于 2020-11-28 14:32...
2.4.1 Definition of Riemann-Stieltjes integral An entirely different way of representing the sampling operation associated with the so-called delta function is to resort to a simple generalisation of the concept of integration itself. Recall that the elementary (or Riemann) theory of the integration...
Riemann-Stieltjes 积分
integral.Themainobjectiveofthischapteris,then,toprovideacarefulintroduc- tiontothetheoryofStieltjes(andhenceRiemann)integration.Ourtreatmentis fairlycomprehensive,butparticularemphasisisgiventotheconnectionbetweenthe LebesgueandStieltjesintegralswithintherealmofprobabilitytheory.Moreover,we ...
integration by partsChoquet-Borel-Stieltjes integralChoquet line integralIn this paper we introduce a new concept of Choquet-Stieltjes integral of f with respect to g on intervals, as a limit of Choquet integrals with respect to a capacity 渭 . For g ( t ) = t , one reduces to the ...
13.2 Denote by S the natural semiring of I subinterval of R . If 渭 is any measure on S , every indefinite integral of 渭 (see Definition 13.2.1) is a function of locally bounded variation (Theorem 13.2.1). The formula for integration by parts holds true for functions of locally ...
There are many instances where the partial summation and integration by parts are regarded as different processes. However, from the general point of view of Stieltjes integration they are exactly the same and it allows one to treat the sum and integral in an essentially similar fashion. Cf. [...