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Hans Adolph Rademacher (; 3 April 1892, Wandsbeck, now
Hamburg-Wandsbek Wandsbek () is an urban quarter in the Wandsbek borough of Hamburg, Germany, and the former city Wandsbek in the Duchy of Holstein. In 2020 the population was 36,671. History Wandsbek was once part of the county ''Stormarn''. Its villages were ...
– 7 February 1969,
Haverford, Pennsylvania Haverford is an unincorporated community located in both Haverford Township in Delaware County, Pennsylvania, United States, and Lower Merion Township in Montgomery County, approximately west of Philadelphia. The Pennsylvania Railroad (PRR) open ...
, USA) was a
German German(s) may refer to: * Germany (of or related to) ** Germania (historical use) * Germans, citizens of Germany, people of German ancestry, or native speakers of the German language ** For citizens of Germany, see also German nationality law **Ge ...
-born American
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, structure, space, models, and change. History On ...
, known for work in
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
and
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
.


Biography

Rademacher received his Ph.D. in 1916 from Georg-August-Universität Göttingen;
Constantin Carathéodory Constantin Carathéodory ( el, Κωνσταντίνος Καραθεοδωρή, Konstantinos Karatheodori; 13 September 1873 – 2 February 1950) was a Greek mathematician who spent most of his professional career in Germany. He made significant ...
supervised his dissertation. In 1919, he became '' privatdozent'' under
Constantin Carathéodory Constantin Carathéodory ( el, Κωνσταντίνος Καραθεοδωρή, Konstantinos Karatheodori; 13 September 1873 – 2 February 1950) was a Greek mathematician who spent most of his professional career in Germany. He made significant ...
at
University of Berlin Humboldt-Universität zu Berlin (german: Humboldt-Universität zu Berlin, abbreviated HU Berlin) is a German public research university in the central borough of Mitte in Berlin. It was established by Frederick William III on the initiative ...
. In 1922, he became an assistant professor at the
University of Hamburg The University of Hamburg (german: link=no, Universität Hamburg, also referred to as UHH) is a public research university in Hamburg, Germany. It was founded on 28 March 1919 by combining the previous General Lecture System ('' Allgemeines Vo ...
, where he supervised budding mathematicians like
Theodor Estermann Theodor Estermann (5 February 1902 – 29 November 1991) was a German-born American mathematician, working in the field of analytic number theory. The Estermann measure, a measure of the central symmetry of a convex set in the Euclidean plan ...
. He was dismissed from his position at the
University of Breslau A university () is an institution of higher (or tertiary) education and research which awards academic degrees in several academic disciplines. Universities typically offer both undergraduate and postgraduate programs. In the United States, th ...
by the
Nazi Nazism ( ; german: Nazismus), the common name in English for National Socialism (german: Nationalsozialismus, ), is the far-right totalitarian political ideology and practices associated with Adolf Hitler and the Nazi Party (NSDAP) in ...
s in 1933 due to his public support of the
Weimar Republic The Weimar Republic (german: link=no, Weimarer Republik ), officially named the German Reich, was the government of Germany from 1918 to 1933, during which it was a constitutional federal republic for the first time in history; hence it is al ...
, and emigrated from Europe in 1934. After leaving Germany, he moved to
Philadelphia Philadelphia, often called Philly, is the largest city in the Commonwealth of Pennsylvania, the sixth-largest city in the U.S., the second-largest city in both the Northeast megalopolis and Mid-Atlantic regions after New York City. Sinc ...
and worked at the
University of Pennsylvania The University of Pennsylvania (also known as Penn or UPenn) is a private research university in Philadelphia. It is the fourth-oldest institution of higher education in the United States and is ranked among the highest-regarded universitie ...
until his retirement in 1962; he held the Thomas A. Scott Professorship of Mathematics at Pennsylvania from 1956 to 1962. Rademacher had a number of well-known students, including George Andrews, Paul T. Bateman,
Theodor Estermann Theodor Estermann (5 February 1902 – 29 November 1991) was a German-born American mathematician, working in the field of analytic number theory. The Estermann measure, a measure of the central symmetry of a convex set in the Euclidean plan ...
and Emil Grosswald.


Research

Rademacher performed research in
analytic number theory In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Diric ...
, mathematical
genetics Genetics is the study of genes, genetic variation, and heredity in organisms.Hartl D, Jones E (2005) It is an important branch in biology because heredity is vital to organisms' evolution. Gregor Mendel, a Moravian Augustinian friar wor ...
, the theory of functions of a real variable, and
quantum theory Quantum theory may refer to: Science *Quantum mechanics, a major field of physics *Old quantum theory, predating modern quantum mechanics * Quantum field theory, an area of quantum mechanics that includes: ** Quantum electrodynamics ** Quantum ...
. Most notably, he developed the theory of
Dedekind sum In mathematics, Dedekind sums are certain sums of products of a sawtooth function, and are given by a function ''D'' of three integer variables. Dedekind introduced them to express the functional equation of the Dedekind eta function. They have su ...
s. In 1937 Rademacher discovered an exact convergent series for the partition function P(n), the number of integer partitions of a number, improving upon Ramanujan's asymptotic non-convergent series and validating Ramanujan's supposition that an exact series representation existed.


Awards and honors

With his retirement from the
University of Pennsylvania The University of Pennsylvania (also known as Penn or UPenn) is a private research university in Philadelphia. It is the fourth-oldest institution of higher education in the United States and is ranked among the highest-regarded universitie ...
, a group of mathematicians provided the seed funding for The Hans A. Rademacher Instructorships, and honored him with an honorary degree as Doctor of Science. Rademacher is the co-author (with
Otto Toeplitz Otto Toeplitz (1 August 1881 – 15 February 1940) was a German mathematician working in functional analysis., reprinted in Life and work Toeplitz was born to a Jewish family of mathematicians. Both his father and grandfather were ''Gymnasiu ...
) of the popular mathematics book ''The Enjoyment of Mathematics'', published in German in 1930 and still in print.


Works

* with
Otto Toeplitz Otto Toeplitz (1 August 1881 – 15 February 1940) was a German mathematician working in functional analysis., reprinted in Life and work Toeplitz was born to a Jewish family of mathematicians. Both his father and grandfather were ''Gymnasiu ...
: ''Von Zahlen und Figuren.'' 1930. 2nd edn. 1933. Springer 2001, . * '' The Enjoyment of Mathematics''. ''Von Zahlen und Figuren'' translated into English by Herbert Zuckerman, Princeton University Press, 1957 * with
Ernst Steinitz Ernst Steinitz (13 June 1871 – 29 September 1928) was a German mathematician. Biography Steinitz was born in Laurahütte (Siemianowice Śląskie), Silesia, Germany (now in Poland), the son of Sigismund Steinitz, a Jewish coal merchant, and ...
''Vorlesungen über die Theorie der Polyeder- unter Einschluss der Elemente der Topologie.'' Springer 1932, 1976. * ''Generalization of the Reciprocity Formula for Dedekind Sums.'' In: ''Duke Math. Journal.'' Vol. 21, 1954, pp. 391–397. * ''Lectures on analytic number theory.'' 1955. * ''Lectures on elementary number theory.'' Blaisdell, New York 1964, Krieger 1977. * with Grosswald: ''Dedekind sums.'' Carus Mathematical Monographs 1972. * ''Topics in analytic number theory.'' ed. Grosswald. Springer Verlag, 1973 (Grundlehren der mathematischen Wissenschaften). * ''Collected papers.'' 2 vols. ed. Grosswald. MIT press, 1974. *''Higher mathematics from an elementary point of view.'' Birkhäuser 1983.


Further reading

* George E. Andrews, David M. Bressoud, L. Alayne Parson (eds.) ''The Rademacher legacy to mathematics.'' American Mathematical Society, 1994. * ''Lexikon bedeutender Mathematiker.'' Deutsch, Thun, Frankfurt am Main, . * Tom Apostol: ''Introduction to Analytical number theory.'' Springer * Tom Apostol: ''Modular functions and Dirichlet Series in Number Theory.'' Springer * Obituary and list of publications.


See also

*
Hadamard transform The Hadamard transform (also known as the Walsh–Hadamard transform, Hadamard–Rademacher–Walsh transform, Walsh transform, or Walsh–Fourier transform) is an example of a generalized class of Fourier transforms. It performs an orthogonal ...
* Rademacher's contour *
Rademacher complexity In computational learning theory (machine learning and theory of computation), Rademacher complexity, named after Hans Rademacher, measures richness of a class of real-valued functions with respect to a probability distribution. Definitions Ra ...
*
Rademacher function In mathematics, the Haar wavelet is a sequence of rescaled "square-shaped" functions which together form a wavelet family or basis. Wavelet analysis is similar to Fourier analysis in that it allows a target function over an interval to be repres ...
* Rademacher–Menchov theorem * Rademacher's series *
Rademacher system Rademacher is an occupational surname of German origin, which means "wheelmaker". It may refer to: People * Arthur Rademacher (1889–1981), Australian football player * Autumn Rademacher (born 1975), American basketball coach * Bill Rademacher (bo ...
*
Rademacher distribution In probability theory and statistics, the Rademacher distribution (which is named after Hans Rademacher) is a discrete probability distribution where a random variate ''X'' has a 50% chance of being +1 and a 50% chance of being -1. A series ( ...
*
Rademacher's theorem In mathematical analysis, Rademacher's theorem, named after Hans Rademacher, states the following: If is an open subset of and is Lipschitz continuous, then is differentiable almost everywhere in ; that is, the points in at which is ''not'' d ...


References


External links

* * 1892 births 1969 deaths 20th-century American mathematicians Mathematics popularizers German emigrants to the United States 20th-century German mathematicians Number theorists Mathematical analysts University of Pennsylvania faculty Mathematicians at the University of Pennsylvania New York University faculty University of Göttingen alumni University of Breslau faculty People from Wandsbek People from Hamburg University of Hamburg faculty Humboldt University of Berlin faculty {{Germany-mathematician-stub