De Bruijn, although still registered as an undergraduate at Leiden, became a full-time Assistant in the Department of Mathematics of the Technological University of Delft in September 1939. He continued to hold this position until June 1944 :-
That period helped him get through a large part of the war without forced labour in Germany (Delft was in the hands of the Germans during the war).In addition to this full-time assistantship, de Bruijn continued his studies at Leiden until 1941. He then studied for a doctorate at the Free University of Amsterdam with Jurjen Koksma as his advisor. He undertook research on algebraic number theory and was awarded a doctorate on 26 March 1943 having submitted his thesis Over modulaire vormen van meer veranderlijken Ⓣ. H Rademacher writes:-
Modular functions of several variables were first associated with algebraic fields by Hilbert and by Blumenthal. The purpose of [de Bruijn's dissertation] is to extend Blumenthal's results in the direction which Hecke and his associates have followed in the theory of modular functions of one variable.Already before he had been awarded his doctorate, de Bruijn had begun publishing papers. For example he published On Steiner-Schläfli's hypocycloid (1940) which took a geometrical approach to ideas published by van der Woude earlier that year. In 1941 he published Ein Satz über schlichte Funktionen Ⓣ followed by Common representative systems of two divisions of an aggregate into classes (1943) which generalised a theorem proved for finite sets by Denes König in 1916 and van der Waerden in 1927, then for infinite sets by König and Valko in 1925. Also in 1943, in addition to his doctoral thesis, he published On the absolute convergence of Dirichlet series, On the number of solutions of the system ..., and Almost periodic multiplicative functions which establishes necessary and sufficient conditions for an arithmetic multiplicative function to be almost periodic in the sense of Harald Bohr.
In June 1944 de Bruijn became a Scientific Associate at Philips Research Laboratories in Eindhoven. Shortly after beginning this employment, he married Elizabeth de Groot on 30 August 1944; they had four children, Jorina Aleida born 19 January 1947, Frans Willem born 13 April 1948, Elisabeth born 24 November 1950, and Judith Elizabeth born 31 March 1963. He began publishing papers on combinatorics relevant to his work during this period such as The problem of optimum antenna current distribution (1946), A combinatorial problem (1946), On the zeros of a polynomial and of its derivative (1946), and A note on van der Pol's equation (1946) :-
His work on combinatorics resulted in influential notions and results of which we mention the de Bruijn-sequences of 1946 and the de Bruijn-Erdős theorem of 1948.He showed the breadth of his research interests with The electrostatic field of a point charge inside a cylinder, in connection with wave guide theory (written jointly with C J Bouwkamp) in 1947. He held this position at Philips Research Laboratories until October 1946 when he was appointed as a Professor in the Department of Mathematics of the Technological University of Delft.
In September 1952 de Bruijn was appointed as Professor in the Department of Mathematics at the University of Amsterdam. He held this position until September 1960, and it is during this period that he published the remarkable text Asymptotic methods in analysis published by the North-Holland Publishing Company in 1958. De Bruijn writes in the Preface:-
This book arose from a course of lectures... Its purpose is to teach asymptotic methods by explaining a number of examples in every detail, so as to suit beginners who seriously want to acquire some technique in attacking asymptotic problems... This book has not been written exclusively for mathematicians, but also for those physicists and engineers who have a certain maturity... This is not an encyclopaedia on asymptotic results. Not even the asymptotic behaviour of Bessel functions can be found in this book. Attention is focussed mainly on methods.Arthur Erdélyi, reviewing the text wrote:-
This is a highly personal, ingenious, stimulating, and in parts even exhilarating book. It has enough original material to appeal to the expert. It would be an excellent book to read in conjunction with a textbook or monograph on asymptotic expansions - if such a one existed: whether it can replace a textbook is a question which every reader will have to answer for himself.A reprinting, without substantial revisions, was published in 1961, then a Russian translation appeared in 1969, followed by a corrected reprint of the third edition by Dover Publications in 1981. In the Preface to this Dover Reprint, de Bruijn writes:-
My somewhat unusual way of presenting a mathematical subject seems to have been appreciated by many, so much so that a second edition was needed comparatively shortly after the first one. Substantial revisions have not been made. A number of minor errors have been corrected.During the summer of 1959 he was Visiting Gauss Professor at the University of Göttingen. In September 1960 he became Professor in the Department of Mathematics at the Technological University of Eindhoven. He held this position until 1 August 1984 when he retired and was made Professor Emeritus. On returning to Eindhoven he renewed his links with the Philips Research Laboratories where he was made a Consultant in 1960, also a position he held until he retired in 1984.
We have already indicated above some of the remarkable breadth in de Bruijn's mathematical interests. He lists his interests (on his website) as: Geometry, Number Theory, Classical and Functional Analysis, Applied Mathematics, Combinatorics, Computer Science, Logic, Mathematical Language, Brain Models. The Preface  records:-
De Bruijn's famous contributions to mathematics include his work on generalized function theory, analytic number theory, optimal control, quasicrystals, the mathematical analysis of games and much more. In each area he approached, he shed a new light and was known for his originality. De Bruijn could rightly assume the motto "I did it my way" as his own motto. And when it came to automating Mathematics, he again did it his way and introduced the highly influential AUTOMATH. In the past decade he has been also working on the theories of the human brain.We should say a little about AUTOMATH mentioned in the quote. In 1968 he published on automating mathematics, then in 1973 he published the 63 page paper AUTOMATH, a language for mathematics. Charles G Morgan explains that the purpose of the paper is:-
... to present a formal language called AUTOMATH. The rules for writing the language are precise enough that a computer may be used to check a body of text for (grammatical) correctness. The author claims that the language is sufficient to express significant portions of mathematics, including meta-theoretical notions concerning inference rules and proofs. The language is so designed that it is incorrect to state a theorem without first "constructing" a proof of the theorem. Thus a computer verification of a body of text translated into AUTOMATH provides automated proof checking. ... The presentation is relatively informal and primarily by example.De Bruijn has received, and continues to receive, many honours for his outstanding mathematical contributions. He lists on his website: Member Royal Netherlands Academy of Sciences since 1957; Invited Speaker at the International Congress of Mathematicians in Nice 1970; Ridder Nederlandse Leeuw (Knight in the Order of the Lion) 1981; Snellius Medal 1985 (This medal of the "Genootschap ter Bevordering van natuur-, genees-, en heelkunde" is awarded only once in 9 years for the whole field of mathematics, natural sciences and medicine. The work on the Automath project was the main motivation for this case); Honorary Member of the Dutch Mathematical Society (Wiskundig Genootschap) since 1988; AKZO Prize 1991; and in 2003 the Lifetime Achievement Award Nederlandse Vereniging v. Theoretische Informatica.
In November 2013 the Royal Netherlands Academy of Arts and Sciences published a special issue of Indagationes Mathematicea: In memory of N.G. (Dick) de Bruijn (1918-2012) .
Article by: J J O'Connor and E F Robertson