Paul du Bois-Reymond

Paul David Gustav du Bois-Reymond.

Paul David Gustav du Bois-Reymond.

Paul David Gustav du Bois-Reymond(2 December 1831 – 7 April 1889) was a Germanmathematicianwho was born inBerlinand died inFreiburg.He was the brother ofEmil du Bois-Reymond.

His thesis was concerned with themechanical equilibriumof fluids. He worked on thetheory of functionsand inmathematical physics.His interests includedSturm–Liouville theory,integral equations,variational calculus,andFourier series.In this latter field, he was able in 1873 to construct a continuous function whose Fourier series is not convergent. Hislemmadefines a sufficient condition to guarantee that a function vanishes almost everywhere.

In a paper of 1875, du Bois-Reymond employed for the first time the method of diagonalization, later associated with the name of Cantor («Über asymptotische Werte, infinitäre Approximationen und infinitäre Auflösungen von Gleichungen», Mathematische Annalen 8: 363-414, doi:10.1007/bf01443187). Du Bois-Reymond also established that atrigonometric seriesthat converges to a continuous function at every point is the Fourier series of this function. He is also associated with thefundamental lemma of calculus of variationsof which he proved a refined version based on that ofLagrange.

Theory of infinitesimals

Paul du Bois-Reymond developed a theory ofinfinitesimals:

The infinitely small is a mathematical quantity and has all its properties in common with the finite […] A belief in the infinitely small does not triumph easily. Yet when one thinks boldly and freely, the initial distrust will soon mellow into a pleasant certainty... A majority of educated people will admit an infinite in space and time, and not just an "unboundedly large". But they will only with difficulty believe in the infinitely small, despite the fact that the infinitely small has the same right to existence as the infinitely large. […]

— Paul du Bois-Reymond,Über die Paradoxen des Infinitär-Calcüls(On the paradoxes of the infinitary calculus), 1877

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)

External links

• Works by or about Paul du Bois-ReymondatInternet Archive
• O'Connor, John J.;Robertson, Edmund F.,"Paul du Bois-Reymond",MacTutor History of Mathematics Archive,University of St Andrews
• Paul du Bois-Reymondat theMathematics Genealogy Project