Paul du Bois-Reymond
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Paul David Gustav du Bois-Reymond (2 December 1831 – 7 April 1889) was a German
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
who was born in
Berlin Berlin ( ; ) is the Capital of Germany, capital and largest city of Germany, by both area and List of cities in Germany by population, population. With 3.7 million inhabitants, it has the List of cities in the European Union by population withi ...
and died in
Freiburg Freiburg im Breisgau or simply Freiburg is the List of cities in Baden-Württemberg by population, fourth-largest city of the German state of Baden-Württemberg after Stuttgart, Mannheim and Karlsruhe. Its built-up area has a population of abou ...
. He was the brother of Emil du Bois-Reymond. His thesis was concerned with the mechanical equilibrium of fluids. He worked on the theory of functions and in
mathematical physics Mathematical physics is the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the de ...
. His interests included Sturm–Liouville theory,
integral equation In mathematical analysis, integral equations are equations in which an unknown function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: f(x_1,x_2,x_3,\ldots,x_n ; u(x_1,x_2 ...
s, variational calculus, and
Fourier series A Fourier series () is an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems ...
. In this latter field, he was able in 1873 to construct a continuous function whose Fourier series is not convergent. His lemma defines a sufficient condition to guarantee that a function vanishes
almost everywhere In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
. In a paper of 1875, du Bois-Reymond employed for the first time the method of diagonalization, later associated with the name of Cantor. Du Bois-Reymond also established that a trigonometric series that converges to a continuous function at every point is the Fourier series of this function. He is also associated with the fundamental lemma of calculus of variations of which he proved a refined version based on that of
Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaFriedrich Stegmann, ''Lehrbuch der Variationsrechnung,'' Kassel 1854, dort werden aber einschränkendere Annahmen gemacht.


Theory of infinitesimals

Paul du Bois-Reymond developed a theory of
infinitesimal In mathematics, an infinitesimal number is a non-zero quantity that is closer to 0 than any non-zero real number is. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referred to the " ...
s:


Writings


Théorie générale des fonctions
(Nice : Impr. niçoise, 1887) (translated in French from the original German by G. Millaud and A. Girot)
De Aequilibrio Fluidorum
(PhD Thesis, 1859)


References


External links

* * * * 1831 births 1889 deaths 19th-century German mathematicians Mathematicians from Berlin People from the Province of Brandenburg Academic staff of the University of Freiburg Academic staff of the University of Tübingen Academic staff of Technische Universität Berlin German people of French descent Mathematicians from the Kingdom of Prussia Mathematicians from the German Empire {{germany-mathematician-stub