convergence of integrals


Similarly as one speaks of convergence of series, one can speak of convergence of integrals, especially of Riemann integrals

∫If⁒(t)⁒𝑑t.

This integralDlmfPlanetmath is convergent, if it exists, and otherwise divergent.  One can also speak of absolute convergence of integrals.

Example.  Study the convergence of the integral

∫12d⁒x(ln⁑x)c (1)

where c is a real constant.

According to the logarithm series, we may write for  1<x<b,  where b is sufficiently close to 1, the estimations

ln⁑(x-1)=x-1+O⁒((x-1)2)=(x-1)⁒[1+O⁒(x-1)]⁒{≀2⁒(x-1),β‰₯12⁒(x-1).

Let  1<a<b. 

1∘.  For  c>1:

∫abd⁒x(ln⁑x)c β‰§βˆ«abd⁒x2c⁒(x-1)c=-12c⁒/x=ab⁑1(c-1)⁒(x-1)c-1
 =12c⁒(c-1)⁒[1(a-1)c-1-1(b-1)c-1]βŸΆβˆžβ€ƒas aβ†’1+

2∘.  For  c=1:

∫abd⁒xln⁑x β‰§βˆ«abd⁒x2⁒(x-1)=12⁒/ab⁑ln⁑(x-1)
 =12⁒[ln⁑(b-1)-ln⁑(a-1)]βŸΆβˆžβ€ƒas aβ†’1+

3∘.  For  c<1:

0<∫abd⁒x(ln⁑x)c β‰¦βˆ«ab2c⁒d⁒x(x-1)c= 2c⁒/x=ab⁑x1-c1-c
 =2c1-c⁒[(b-1)1-c-(a-1)1-c]⟢2c1-c⁒(b-1)1-c as aβ†’1+

Consequently, the integral ∫abd⁒x(ln⁑x)c, and thus also (1), converges if and only if  c<1.

Title convergence of integrals
Canonical name ConvergenceOfIntegrals
Date of creation 2013-03-22 18:59:51
Last modified on 2013-03-22 18:59:51
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 9
Author pahio (2872)
Entry type Example
Classification msc 40A10
Related topic UniformConvergenceOfIntegral
Related topic LogarithmicIntegral2
Related topic ListOfImproperIntegrals
Related topic SubstitutionNotation
Related topic ONotation
Defines convergent integral
Defines divergent integral