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 zu verstehen?
Um dieses festzusetzen, nehmen wir zwischen und der Grösse nach auf einander folgend, eine Reihe von Werthen an und bezeichnen der Kürze wegen durch , durch durch und durch einen positiven ächten Bruch. Es wird alsdann der Werth der Summe
von der Wahl der Intervalle und der Grössen abhängen. Hat sie nun die Eigenschaft, wie auch und gewählt werden mögen, sich einer festen Grenze unendlich zu nähern, sobald sämmtliche unendlich klein werden, so heisst dieser Werth .
In Birkhoff’s book:
First of all: What is to be understood by ?
In order to establish this, we take the sequence of values lying between and and ordered by size, and, for brevity, denote by , by by , and proper positive fractions by . Then the value of the sum
will depend on the choice of the intervals and the quantities . If it has the property that, however the and the may be chosen, it tends to a fixed limit as soon as all the become infinitely small, then this value is called .
(Of course, in modern presentations, we use instead of , and say that the approach rather than become infinitely small. In fact, we tend to call the collection of data , a tagged partition of , and call the maximum of the the mesh or norm of the partition.)
This set is due Friday, December 6, at the beginning of lecture.
Newton’s method was introduced by Newton on De analysi in 1669. It was originally restricted to polynomials; his example in Methodus fluxionum was the cubic equation
Raphson simplified its description in 1690. The modern presentation, in full generality, is due to Simpson in 1740. Here, we are mostly interested in the dynamics of Newton’s method on polynomials.
Weierstrass function from 1872 is the function defined by
Weierstrass showed that if
- is an odd positive integer, and
then is a continuous nowhere differentiable function. Hardy proved in 1916 that one can relax the conditions on to
- , and
Here, I just want to show some graphs, hopefully providing some intuition to help understand why we expect to be non-differentiable. The idea is that the cosine terms ensure that the partial sums , though smooth, have more and more “turns” on each interval as increases, so that in the limit, has “peaks” everywhere. Below is an animation (produced using Sage) comparing the graphs of for (and ), for and , showing how the bends accumulate. (If the animations are not running, clicking on them solves the problem. As far as I can see, they do not work on mobiles.)
Following a theme from two years ago, we will have a final project for this course, due Wednesday, December 18, by noon, but feel free (and encouraged) to turn it in earlier. (As discussed in lecture, the project is voluntary for some of you. Contact me if you are not sure whether it is required or voluntary for you.)
There are many excellent sources on the topic of continuous nowhere differentiable functions. Johan Thim’s Master thesis, written under the supervision of Lech Maligranda, is available online, here, but feel free to use any other sources you find relevant.
Please choose an example of a continuous nowhere differentiable function, either from Thim’s thesis or elsewhere, and write (better yet, type) a note on who it is due to and what the function is, together with complete proofs of continuity and nowhere differentiability. Though not required, feel free (and encouraged) to add additional information you consider relevant for context.
(For an example of what I mean by relevant additional information: Weierstrass function is where , is an odd positive integer, and . It may be interesting to add a discussion of precisely what conditions are needed from to ensure (continuity and) nowhere differentiability; Weierstrass original requirements are more restrictive than necessary. For another example, Schoenberg functions, discussed in Thim’s thesis, give a natural example of a space filling curve, so consider including a proof of this fact.)
Please take this project very seriously (in particular, do not copy details from books or papers, I want to see your own version of the details as you work through the arguments). Feel free to ask for feedback as you work on it; in fact, asking for feedback is a good idea. Do not wait until the last minute. At the end, it would be nice to make at least some of the notes available online, please let me know when you turn it in whether you grant me permission to host your note on this blog.
Here is a list of the projects I posted on the blog, from last time:
- Katsuura function, by Erron Kearns.
- Faber functions, by Shehzad Ahmed.
- Schoenberg functions, by Jeremy Ryder.
Contact me (by email) as soon as you have chosen the example you will work on, to avoid repetitions; I will add your name and the chosen example to the list below as I hear from you.
List of projects:
- Joe Busick: Katsuura function.
- Paul Carnig: Darboux function.
- Joshua Meier: A variant of Koch’s snowflake.
- Paul Plummer: Lynch function.
- Veronica Schmidt: McCarthy function.