History of the integral

1. Feb 15, 2006

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

2. Feb 15, 2006

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 xi to xi+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.

3. Feb 16, 2006

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.

4. Feb 16, 2006

Hurkyl

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

5. Feb 16, 2006

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.

6. Feb 16, 2006

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.

7. Feb 16, 2006

arildno

Last edited: Feb 16, 2006