Riemann on Riemann sums

Though Riemann sums had been considered earlier, at least in particular cases (for example, by Cauchy), the general version we consider today was introduced by Riemann, when studying problems related to trigonometric series, in his paper Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe. This was his Habilitationsschrift, from 1854, published posthumously in 1868.

Riemann’s papers (in German) have been made available by the Electronic Library of Mathematics, see here. The text in question appears in section 4, Ueber den Begriff eines bestimmten Integrals und den Umfang seiner Gültigkeit. The translation below is as in

  • A source book in classical analysis. Edited by Garrett Birkhoff. With the assistance of Uta Merzbach. Harvard University Press, Cambridge, Mass., 1973. MR0469612 (57 #9395).

Also zuerst: Was hat man unter \displaystyle \int_a^b f(x) \, dx zu verstehen?

Um dieses festzusetzen, nehmen wir zwischen a und b der Grösse nach auf einander folgend, eine Reihe von Werthen x_1, x_2,\ldots, x_{n-1} an und bezeichnen der Kürze wegen x_1 - a durch \delta_1, x_2 - x_1 durch \delta_2,\ldots, b - x_{n-1} durch \delta_n und durch \varepsilon einen positiven ächten Bruch.  Es wird alsdann der Werth der Summe

\displaystyle S = \delta_1 f(a + \varepsilon_1 \delta_1) + \delta_2 f(x_1 + \varepsilon_2 \delta_2) + \delta_3 f(x_2 + \varepsilon_3 \delta_3) +\cdots \displaystyle +\delta_n f(x_{n-1} +\varepsilon_n \delta_n)

von der Wahl der Intervalle \delta und der Grössen \varepsilon abhängen.  Hat sie nun die Eigenschaft, wie auch \delta und \varepsilon gewählt werden mögen, sich einer festen Grenze A unendlich zu nähern, sobald sämmtliche \delta unendlich klein werden, so heisst dieser Werth \displaystyle \int_a^b f(x) \, dx.

In Birkhoff’s book:

First of all: What is to be understood by \displaystyle \int_a^b f(x)\,dx?

In order to establish this, we take the sequence of values x_1,x_2,\ldots, x_{n-1} lying between a and b and ordered by size, and, for brevity, denote x_1 - a by \delta_1, x_2 - x_1 by \delta_2,\ldots, b - x_{n-1} by \delta_n, and proper positive fractions by \varepsilon_i. Then the value of the sum

\displaystyle S = \delta_1 f(a + \varepsilon_1 \delta_1) + \delta_2 f(x_1 + \varepsilon_2 \delta_2) + \delta_3 f(x_2 + \varepsilon_3 \delta_3) +\cdots \displaystyle +\delta_n f(x_{n-1} +\varepsilon_n \delta_n)

will depend on the choice of the intervals \delta_i and the quantities \varepsilon_i. If it has the property that, however the \delta_i and the \varepsilon_i may be chosen, it tends to a fixed limit A as soon as all the \delta_i become infinitely small, then this value is called \displaystyle \int_a^b f(x) \, dx.

(Of  course, in modern presentations, we use \Delta_i instead of \delta_i, and say that the \delta_i approach 0 rather than become infinitely small. In fact, we tend to call the collection of data x_1,\dots,x_{n-1}, \varepsilon_1,\dots,\varepsilon_n a tagged partition of {}[a,b], and call the maximum of the x_{i+1}-x_i the mesh or norm of the partition.)

One Response to Riemann on Riemann sums

  1. […] 11. Section 7.1. Riemann sums. Homework: 7.1: 2, 4, 6, […]

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