The Riemann-Stieltjes Integral IV
Let’s do one more easy application of the Riemann-Stieltjes integral. We know from last Friday that when our integrator is continuously differentiable, we can reduce to a Riemann integral:
![\displaystyle\int\limits_{\left[a,b\right]}f(x)d\alpha(x)=\int\limits_{\left[a,b\right]}f(x)\alpha'(x)dx](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cint%5Climits_%7B%5Cleft%5Ba%2Cb%5Cright%5D%7Df%28x%29d%5Calpha%28x%29%3D%5Cint%5Climits_%7B%5Cleft%5Ba%2Cb%5Cright%5D%7Df%28x%29%5Calpha%27%28x%29dx&bg=e6e6e6&fg=333333&s=0 "\displaystyle\int\limits_{\left[a,b\right]}f(x)d\alpha(x)=\int\limits_{\left[a,b\right]}f(x)\alpha'(x)dx")
So where else have we seen derivatives as factors in integrands? Right! integration by parts! Here our formula says that
We can rewrite this using Riemann-Stieltjes integrals as
![\displaystyle\int\limits_{\left[a,b\right]}f(x)dg(x)+\int\limits_{\left[a,b\right]}g(x)df(x)=f(b)g(b)-f(a)g(a)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cint%5Climits_%7B%5Cleft%5Ba%2Cb%5Cright%5D%7Df%28x%29dg%28x%29%2B%5Cint%5Climits_%7B%5Cleft%5Ba%2Cb%5Cright%5D%7Dg%28x%29df%28x%29%3Df%28b%29g%28b%29-f%28a%29g%28a%29&bg=e6e6e6&fg=333333&s=0 "\displaystyle\int\limits_{\left[a,b\right]}f(x)dg(x)+\int\limits_{\left[a,b\right]}g(x)df(x)=f(b)g(b)-f(a)g(a)")
So if 
and 
are both continuously differentiable, this formula gives back our rule for integration by parts. But we can prove this without making those assumptions. In fact, we just need to assume that one of the two integrals exists, and the existence of the other one (and the formula) will follow.
Let’s assume that ![\int_{\left[a,b\right]}f,dg](https://s0.wp.com/latex.php?latex=%5Cint_%7B%5Cleft%5Ba%2Cb%5Cright%5D%7Df%2Cdg&bg=e6e6e6&fg=333333&s=0 "\int_{\left[a,b\right]}f,dg")
exists. That is, for every 
there is some tagged partition 
so that for every finer partition 
we have
![\displaystyle\left|\int\limits_{\left[a,b\right]}f(x)dg(x)-f_{g,x}\right|<\epsilon](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cleft%7C%5Cint%5Climits_%7B%5Cleft%5Ba%2Cb%5Cright%5D%7Df%28x%29dg%28x%29-f_%7Bg%2Cx%7D%5Cright%7C%3C%5Cepsilon&bg=e6e6e6&fg=333333&s=0 "\displaystyle\left|\int\limits_{\left[a,b\right]}f(x)dg(x)-f_{g,x}\right|<\epsilon")
Now let’s take any partition finer than and use it to set up the Rieman-Stieltjes sum
We can also use this partition to rewrite 
as
So subtracting the one from the other we find
But this is a Riemann-Stieltjes sum 
for the partition 
we get by throwing together all the 
and 
as partition points, and using as tags. This is a finer partition than , and so we see that
![\displaystyle\left|\left(A-\int\limits_{\left[a,b\right]}f(x)dg(x)\right)-g_{f,x}\right|=\left|\int\limits_{\left[a,b\right]}f(x)dg(x)-\left(A-g_{f,x}\right)\right|<\epsilon](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cleft%7C%5Cleft%28A-%5Cint%5Climits_%7B%5Cleft%5Ba%2Cb%5Cright%5D%7Df%28x%29dg%28x%29%5Cright%29-g_%7Bf%2Cx%7D%5Cright%7C%3D%5Cleft%7C%5Cint%5Climits_%7B%5Cleft%5Ba%2Cb%5Cright%5D%7Df%28x%29dg%28x%29-%5Cleft%28A-g_%7Bf%2Cx%7D%5Cright%29%5Cright%7C%3C%5Cepsilon&bg=e6e6e6&fg=333333&s=0 "\displaystyle\left|\left(A-\int\limits_{\left[a,b\right]}f(x)dg(x)\right)-g_{f,x}\right|=\left|\int\limits_{\left[a,b\right]}f(x)dg(x)-\left(A-g_{f,x}\right)\right|<\epsilon")
whenever is a partition finer than . This shows that the Riemann-Stieltjes integral of with respect to exists, and has the value we want.
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March 4, 2008 -
Posted by John Armstrong |
Analysis, Calculus
[…] is similar to the formula for integration by parts, and is referred to as Abel’s partial summation formula. In particular, it tells us that the […]
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