During some recent work on the computation of the inverse Langevin function I ran into trouble trying to generate a highly-accurate minimax rational approximation with a variant of the Remez algorithm, encountering failures to converge and generation of degenerate solutions that fail to preserve minimax properties. That prompted me to look around for alternative ways of generating rational approximations, and one paper I came across during my literature search is:

C. M. Lee and F. D. K. Roberts, "A Comparison of Algorithms for Rational $l_{\infty}$ Approximation", Mathematics of Computation, Vol. 27, No. 121, Jan. 1973, pp. 111-121

This mentions Maehly's Method as one possible alternative to the Remez algorithm, and I recalled coming across this years ago in this publication:

E.G. Kogbetliantz, "Generation of Elementary Functions". In: Anthony Ralston and Herbert S. Wilf (eds.), Mathematical Methods for Digital Computers, New York: Wiley 1960, pp. 7-35

I also noted that Maehly's name occurs on the author list of a standard reference for approximations to various elementary functions that is still being consulted today despite its age:

John F. Hart, E.W. Cheney, Charles L. Lawson, Hans J. Maehly, Charles K. Mesztenyi, John R. Rice, Henry G. Thacher, Jr., and Christoph Witzgall, Computer Approximations. New York: Wiley 1968

Google Scholar showed only half a dozen publications by Hans J. Maehly, prompting the titular question.


1 Answer 1


An initial Google search returned one serendipitous result that immediately put me on the right track because it mentioned that Maehly had immigrated to the US from Switzerland shortly after receiving a Ph.D. from ETH Zurich, and had died in 1961 at a young age:

Walter Gautschi, "Schweizer Expats in den USA". Notes from a talk given in January 12, 2016 at ETH Zürich:

Die Liste der angeführten Expats ist natürlich bei weitem nicht vollständig und beschränkt sich ausschliesslich auf Leute aus meinem Bekanntenkreis. Dazu gehören auch zwei damals vielversprechende junge Mathematiker, Hans J. Maehly und Andreas H. Schopf, die kurz nach deren Promovierung an der ETH nach den USA ausgewandert sind, aber dort frühzeitig starben, Schopf 1959 und Maehly 1961.

Reversing the common transcription of the umlaut 'ä' as "ae", I found Maehly's dissertation right away at the ETH digital repository. Maehly received a Ph.D. in physics from Eidgenössische Technische Hochschule (ETH) in Zurich in 1951 with a dissertation in mathematical physics, advised by Paul Scherrer:

Hans Jakob Mähly, Methoden zur genäherten Berechnung von Eigenwerten elastischer Schwingungen anisotroper Körper, Ph.D. dissertation, ETH Zürich 1951

This matches the information at the Mathematics Genealogy Project. There are some brief biographical notes at the end of the thesis: Maehly was born on June 25, 1920 in Basel. He graduated from high school in the spring of 1939. After one semester at the University of Basel he transferred to the department of mathematics and physics at ETH. In 1944 he received his diplom degree as an experimental physicist; his diplom thesis was on the elastic properties of seignette-electric potassium phosphate.

While at ETH, Maehly published two papers. The first one lays a foundation for the Lehmann-Maehly method of determining bounds of eigenvalues (Lehmann independently published on this). The second details a root finding method sometimes referred to as Maehly's Procedure.

H. J. Maehly, "Ein neues Variationsverfahren zur genäherten Berechnung der Eigenwerte hermitescher Operatoren". Helvetica Physica Acta, Vol. 25, No. 5, 1952, pp. 547-568

Hans J. Maehly, "Zur iterativen Auflösung algebraischer Gleichungen". Zeitschrift für Angewandte Mathematik und Physik, Vol. 5, No. 3, May 1954, pp. 260-263

Maehly's institutional affiliation for the first paper is given as ETH's Institute of Physics, while for the second it is stated as the Institute of Applied Mathematics of ETH. Going forward, Maehly appears to have worked on mathematical problems exclusively, all of them in the field of numerical analysis. He was also a pioneer in numerical computation using computers. In the surviving usage records for ETH's Z4 computer, his name occurs three times. Herbert Bruderer, "Wofür wurde der Relaisrechner Z4 von Konrad Zuse an der ETH Zürich verwendet?" ETH Zürich, Departement Informatik, Zürich 2011:

200 hours: partial differential equations of order 4 (1951, 1952)
50 hours: root finding in polynomials; context: quantum mechanical computations for aromatic compounds (1953, 1954)
170 hours: solving algebraic equations (1953)

At some time in the mid-1950s, Maehly left Switzerland to work at Princeton. I have not been able to establish the exact date or his specific motivation. The earliest reference I could find to Maehly's presence at Princeton is a reference in Kurt Spielberg, "Efficient Continued Fraction Approximations To Elementary Functions." Mathematics of Computation, Vol. 15, No. 76, 1961, pp. 409-417:

H. J. Maehly, Monthly Progress Report (unpublished), February 1956

From the institutional affiliations listed in his publications, he appears to have staid at Princeton until sometime in 1959, leaving for Syracuse University. A letter to the editor in Communications of the ACM, Vol. 3, No. 3, March 1960, p. A11 gives his affiliation as Department of Mathematics, Syracuse University. At Princeton, he appears to have had continued intense involvement with computers, as evidenced by the following note to department heads dated September 17, 1957:

Princeton University has recently taken over operation of the electronic digital computer previously operated by the Institute for Advanced Study and known as the MANIAC. This computer is now available for use by members of the academic community. [...] Dr. Hans Maehly is in charge of the computer and should be consulted if there are any questions concerning its use;

I assembled the following list of Maehly's publications during his time in the United States:

H. J. Maehly, "Rational approximations for transcendental functions". In Proceedings of the International Conference on Information Processing, UNESCO, June 1959, pp. 57-62. London: Butterworth 1960.

H. J. Maehly, Approximations for the CDC 1604. Control Data Corp 1960

Hans J. Maehly, "Methods for Fitting Rational Approximations, Part I: Telescoping Procedures for Continued Fractions". Journal of the ACM, Vol. 7, No. 2, April 1960, pp. 150-162

H. Maehly and Ch. Witzgall, "Tschebyscheff-Approximationen in kleinen Intervallen I: Approximation durch Polynome". Numerische Mathematik, Vol. 2, 1960, pp. 142-150

H. Maehly and Ch. Witzgall, "Tschebyscheff-Approximationen in kleinen Intervallen II: Stetigkeitssätze für gebrochen rationale Approximationen". Numerische Mathematik, Vol. 2, 1960, pp. 293-307

Hans J. Maehly, "Numerical solution of a certain transcendental equation involving exponentials (a remark on a paper by J. R. Rice)". Journal of the Society for Industrial and Applied Mathematics, Vol. 10, No. 1, March 1962, pp. 30-34

Hans J. Maehly, "Methods for fitting rational approximations, Parts II and III". Journal of the ACM, Vol. 10, No. 3, July 1963, pp. 257-277

The preface to the last publication listed, written by Maehly's colleague Christoph Witzgall, informs the reader that he passed away unexpectedly in 1961, shorty after moving from Syracuse to Argonne National Laboratory:

Dr. Hans J. Maehly died on the 16th of November, 1961. Just six weeks before his untimely death, Dr. Maehly had joined the Applied Mathematics Division of Argonne National Laboratory.

Multiple sources indicate that Maehly's sudden death was due to a heart attack:

Thomas Haigh, "An interview with W. J. Cody", December 2005, 53 pp.:

As a matter of fact, when I first started working in that area I discovered a little belatedly that we had one member of our division who was an expert on that, that was Hans Maehly, who had come over from Princeton. I say I discovered belatedly, because he was only there about six months and died of a heart attack in the parking lot, so I knew him but I didn’t know what he was working on, and he didn’t know what I was working on.

Richard Hertz, "Interview with Forman Acton", 1971, 48 pp.

Hans did a couple of pieces of research in numerical methods sorts of things, rather nice, before he departed for Syracuse, if I remember rightly. Then he went from there to Argonne Labs and he’d been at Argonne about two weeks when he died.

"Christoph Witzgall", Journal of Research of the National Institute of Standards and Technology, Vol. 111, No. 2, March-April 2006:

Edmonds recruited Chris to join NBS, which he did in October 1962 during the Cuban Missile Crisis. That summer he had spent at Argonne National Laboratory, editing papers of Maehly, who had died there of a heart attack.

It seemed a bit unusual that Maehly would be a co-author of the book Computer Approximations given that it was published seven years after his death. One of the co-authors explained:

Thomas Haigh, "An interview with Charles L. Lawson, Nov. 2004, 103 pp.

The seven were J. F Hart, E. W. Cheney, C. L. Lawson, C. K. Mesztenyi, J. R. Rice, H. C. Thacher, Jr. and C. Witzgall. [...] An eighth person, Hans J. Maehly, is listed along with these seven as a coauthor of the book because it was felt his work on this issue had already been very significant and he died at a young age just before work on the book began. The book, Computer Approximations, was published by John Wiley in 1968.

  • $\begingroup$ There is a letter from Maehly here (Comm. ACM **3**(3) March 1960, if that's of any use. It gives his affiliation as Dept. Math., Syracuse Univ. $\endgroup$
    – Michael E2
    Jun 19 at 3:41

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