Who was Stieltjes and how did he contribute to the Riemann-Stieltjes integral?

[quasar987]

I'm curious about the history of the integral.

I believe it was Leibniz who introduced the symbol $\int$. But what did that meant for him besides anti-derivative (if anything)? For I am told it is Riemann who in a paper, introduced the know definition of the Riemann-Stieljes' integral in terms of partitions, upper and lower integral, and probably also the Riemann sum caracterisation. Some questions: who's Stieljes?Why is his name linked to that of Riemman? Who discovered the fundamental theorem?

So as you can see I'm very confused about all this. Please, tell me what's what? thx

 

[HallsofIvy]

I don't know about the history but the "Riemann-Stieljes" integral is different from the "Riemann" integral. The Riemann integral, the thing you learn in basic calculus, is derived using the Riemann sums with $\Delta x$ the length of the interval x_i to x_i+1: $\Delta x= x_{i+1}- x_i$. The Riemann-Stieljes integral uses $\Delta x= \alpha(x_{i+1})- \alpha(x_i)$ where $\alpha$ can be any increasing function. In particular, if $\alpha$ is a step function the Riemann-Stieljes integral is a sum.

 

[fourier jr]

the stieljes integral is the one where alpha(x) is any increasing function; the riemann integral is the special case where alpha(x)=x.

 

[Hurkyl]

Doesn't alpha have to be continuous from the right?

 

[shmoe]

alpha doesn't have to be increasing or continuous from the right. The riemann-stieltjes sum makes sense when f and alpha are bounded. The integral may or may not exist though, there are various theorems on existence that I can't usually remember, like f continuous and alpha of bounded variation will do it, but this isn't required.

I'm useless as far as the history goes though, sorry.

 

[Galileo]

I hear many people calling Thomas Johannes Stieltjes Stieljes, but his real name was Stieltjes.
It may help when you seach for info on him. Many use Stieljes and I don't know why or how, I don't think they're different persons. Looking for Stieltjes will return much more results.

 

[arildno]

Here's a biography of Stieltjes:
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Stieltjes.html

He introduced the Riemann-Stieltjes integral in a work on continued fractions, dealing in particular with the 2moment problem".