# Lothar Collatz

### Born: 6 July 1910 in Arnsberg, Westphalia, Germany

Died: 26 September 1990 in Varna, Bulgaria

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**Lothar Collatz**, like most German students of his time, studied at a number of different universities. He entered the University of Greifswald in 1928, moving to Munich, then to Göttingen, and finally to Berlin where he studied for his doctorate under Alfred Klose. Collatz was awarded his doctorate in 1935 for his dissertation

*Das Differenzenverfahren mit höherer Approximation für lineare Differentialgleichunge*Ⓣ. In [8] Meinardus and Nürnberger write:-

Many will know the name of Collatz today because of the "Collatz problem". In many ways it might seem a pity that a mathematician who has produced so much important and fundamental work should be most remembered for a novelty, yet this problem has intrigued mathematician ever since he proposed it in 1937. The Collatz problem is simple to state. Define a functionHe often told how much he had been impressed by the lectures of Hilbert, Courant, von Mises, Schur, and other famous mathematicians of that period. He was convinced that mathematics and mathematicians had a responsibility to apply their results to, and be motivated by, real world phenomena. he never wearied of fighting for this conviction.

*f*on the positive integers by

*f*(

*n*) = 3

*n*+1 if

*n*is odd;

*f*(

*n*) =

*n*/2 if

*n*is even.

*m*define a sequence by putting

*a*(1) =

*m*and, for

*i*≥ 1,

*a*(

*i*+1) =

*f*(

*a*(

*i*)).

*m*, the sequence

*a*(

*i*) always reaches 1? The problem remains unsolved, but before you try a few small numbers yourself looking for a counterexample, let us say that the conjecture has been verified for all numbers

*m*up to about 10

^{14}.

In 1943 Collatz was appointed to a professorship at the Technical University of Hanover. After holding this position for nine years, in 1952 he moved to the University of Hamburg. There he founded the Institute of Applied Mathematics.

MathSciNet lists 238 items under Collatz' name. Most of these publications are in numerical analysis but those range through almost every area within the subject. Among his early papers are

*Genäherte Berechnung von Eigenwerten*Ⓣ (1939) in which he considers various methods of approximating characteristic values,

*Das Hornersche Schema bei komplexen Wurzeln algebraischer Gleichungen*Ⓣ (1940) in which he presents a more efficient way of using Horner's method to approximate the complex roots of an algebraic equation, and

*Schrittweise Näherungen bei Integralgleichungen und Eigenwertschranken*Ⓣ (1940) in which inequalities between the eigenvalues of certain integral equations are studied.

Collatz began publishing important books early in his career.

*Eigenwertprobleme und ihre numerische Behandlung*Ⓣ (1945) contains three parts, the first containing a collection of practical applications which lead to boundary value problems for ordinary and partial differential equations. The second part provides the theory of boundary value problems while the final part provides numerical method to solve these. This was followed by

*Numerische Behandlung von Differentialgleichungen*Ⓣ (1951) which provides a comprehensive text on numerical methods for solving differential equations. Collatz was keen to provide methods for scientists as in

*Differentialgleichungen für Ingenieure: Eine Einführung*Ⓣ (1960):-

An interesting combining of two areas was presented inThis small book gives a wealth of information on differential equations. ... The concise formulation is clear and precise, and it will appeal to engineers on account of its many graphic examples.

*Functional analysis and numerical mathematics*(1966), which was an English translation of a German book published two years earlier. A S Householder writes:-

A very different area was covered inIt seems strange that this book should be the first of its kind, since it hardly needs to be said that "numerical mathematics" must draw heavily from functional analysis. Nevertheless, in an unnecessarily modest preface, the author disclaims any intention of writing a textbook on either functional analysis or numerical mathematics, offering the book instead as illustrating the artificiality of any separation of applied from pure mathematics. In spite of the disclaimer, the book could be used as a basis for a rather extensive course on functional analysis for numerical analysts, and the footnotes and the bibliography provide for a considerable amount of collateral reading.

*Optimierungsaufgaben*Ⓣ (1966) (written jointly with W Wetterling). This is an introductory textbook which introduces students to linear programming, two-person-zero-sum games, quadratic programming, and convex programming. The book

*Aufgaben aus der Angewandten Mathematik*Ⓣ (1972) (with J Albrecht) provides a collection of problems (with their solutions) on the solution of equations and systems of equations, interpolation, quadrature, approximation, and harmonic analysis.

*Approximationstheorie. Tschebyscheffsche Approximation mit Anwendungen*Ⓣ (1973) (with W Krabs) is described by J L Ullman as follows:-

[Later texts by Collatz includeT]he total effect is to provide a stimulating introduction to the subject to people with both pure and applied inclinations, and at the same time, providing a good primary or secondary reference for an advanced undergraduate course, or a beginning research seminar in approximation theory and/or numerical analysis.

*Optimization problems*(1975) and

*Differential equations*(1986), the second of these being an English translation of an earlier German book.

Collatz received many honours for his contributions including election to the German Academy of Scientists Leopoldina, the academy at Bologna and that at Modena. He was made an honorary member of the Hamburg Mathematical Society and given honorary degrees by the University of São Paulo, the Technical University of Vienna, the University of Dundee in Scotland, Brunel University in England, the Technical University of Hanover, and the Technical University of Dresden.

Meinardus and Nürnberger write [6]:-

He died in Varna, Bulgaria, while attending a mathematics conference on Computer Arithmetic.Professor Collatz was a truly wonderful individual. He was modest in his behaviour, and ever amiable and helpful.

**Article by:** *J J O'Connor* and *E F Robertson*

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