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Maxime Bôcher (August 28, 1867 – September 12, 1918) was an 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, mathematical structure, structure, space, Mathematica ...
who published about 100 papers on differential equations,
series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used i ...
, and
algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
. He also wrote elementary texts such as ''Trigonometry'' and ''Analytic Geometry''. Bôcher's theorem, Bôcher's equation, and the
Bôcher Memorial Prize The Bôcher Memorial Prize was founded by the American Mathematical Society in 1923 in memory of Maxime Bôcher with an initial endowment of $1,450 (contributed by members of that society). It is awarded every three years (formerly every five yea ...
are named after him.


Life

Bôcher was born in
Boston Boston is the capital and most populous city in the Commonwealth (U.S. state), Commonwealth of Massachusetts in the United States. The city serves as the cultural and Financial centre, financial center of New England, a region of the Northeas ...
,
Massachusetts Massachusetts ( ; ), officially the Commonwealth of Massachusetts, is a U.S. state, state in the New England region of the Northeastern United States. It borders the Atlantic Ocean and the Gulf of Maine to its east, Connecticut and Rhode ...
. His parents were Caroline Little and Ferdinand Bôcher. Maxime's father was professor of modern languages at the
Massachusetts Institute of Technology The Massachusetts Institute of Technology (MIT) is a Private university, private research university in Cambridge, Massachusetts, United States. Established in 1861, MIT has played a significant role in the development of many areas of moder ...
when Maxime was born, and became Professor of French at
Harvard University Harvard University is a Private university, private Ivy League research university in Cambridge, Massachusetts, United States. Founded in 1636 and named for its first benefactor, the History of the Puritans in North America, Puritan clergyma ...
in 1872. Bôcher received an excellent education from his parents and from a number of public and private schools in Massachusetts. He graduated from the Cambridge Latin School in 1883. He received his first degree from Harvard in 1888. At Harvard, he studied a wide range of topics, including
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
,
Latin Latin ( or ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken by the Latins (Italic tribe), Latins in Latium (now known as Lazio), the lower Tiber area aroun ...
,
chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a physical science within the natural sciences that studies the chemical elements that make up matter and chemical compound, compounds made of atoms, molecules a ...
, philosophy, zoology,
geography Geography (from Ancient Greek ; combining 'Earth' and 'write', literally 'Earth writing') is the study of the lands, features, inhabitants, and phenomena of Earth. Geography is an all-encompassing discipline that seeks an understanding o ...
,
geology Geology (). is a branch of natural science concerned with the Earth and other astronomical objects, the rocks of which they are composed, and the processes by which they change over time. Modern geology significantly overlaps all other Earth ...
, meteorology, Roman art, and
music Music is the arrangement of sound to create some combination of Musical form, form, harmony, melody, rhythm, or otherwise Musical expression, expressive content. Music is generally agreed to be a cultural universal that is present in all hum ...
. Bôcher was awarded many prestigious prizes, which allowed him to travel to Europe to do research. The
University of Göttingen The University of Göttingen, officially the Georg August University of Göttingen (, commonly referred to as Georgia Augusta), is a Public university, public research university in the city of Göttingen, Lower Saxony, Germany. Founded in 1734 ...
was then the leading mathematics university, and he attended there lectures by
Felix Klein Felix Christian Klein (; ; 25 April 1849 – 22 June 1925) was a German mathematician and Mathematics education, mathematics educator, known for his work in group theory, complex analysis, non-Euclidean geometry, and the associations betwe ...
,
Arthur Moritz Schoenflies Arthur Moritz Schoenflies (; 17 April 1853 – 27 May 1928), sometimes written as Schönflies, was a German mathematician, known for his contributions to the application of group theory to crystallography, and for work in topology. Schoenflies ...
,
Hermann Schwarz Karl Hermann Amandus Schwarz (; 25 January 1843 – 30 November 1921) was a German mathematician, known for his work in complex analysis. Life Schwarz was born in Hermsdorf, Silesia (now Sobieszów, Poland). In 1868 he married Marie Kummer ...
,
Issai Schur Issai Schur (10 January 1875 – 10 January 1941) was a Russian mathematician who worked in Germany for most of his life. He studied at the Humboldt University of Berlin, University of Berlin. He obtained his doctorate in 1901, became lecturer i ...
and
Woldemar Voigt Woldemar Voigt (; 2 September 1850 – 13 December 1919) was a German mathematician and physicist. Biography Voigt was born in Leipzig, and died in Göttingen. He was a student of Franz Ernst Neumann. Voigt taught at the Georg August Universi ...
. He was awarded a doctorate in 1891 for his dissertation ''Über die Reihenentwicklungen der Potentialtheorie'' ( German for "On the Development of the Potential Function into Series"); he was encouraged to study this topic by Klein. He received a Göttingen university prize for this work. Bocher was elected to the
American Academy of Arts and Sciences The American Academy of Arts and Sciences (The Academy) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, and other ...
in 1899, the United States
National Academy of Sciences The National Academy of Sciences (NAS) is a United States nonprofit, NGO, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the ...
in 1909, and the
American Philosophical Society The American Philosophical Society (APS) is an American scholarly organization and learned society founded in 1743 in Philadelphia that promotes knowledge in the humanities and natural sciences through research, professional meetings, publicat ...
in 1916. In Göttingen he met Marie Niemann, and they were married in July 1891. They had three children, Helen, Esther, and Frederick. He returned with his wife to Harvard where he was appointed as an instructor. In 1894 he was promoted to assistant professor, due to his impressive record. He became a full professor of mathematics in 1904. He was president of the
American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
from 1908 to 1910. Although he was only 46 years old, there were already signs that his weak health was failing. He died at his
Cambridge Cambridge ( ) is a List of cities in the United Kingdom, city and non-metropolitan district in the county of Cambridgeshire, England. It is the county town of Cambridgeshire and is located on the River Cam, north of London. As of the 2021 Unit ...
home after suffering a prolonged illness.


Bôcher's theorem

Bôcher's theorem states that the finite zeros of the
derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
r'(z) of a non- constant
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
r(z) that are not multiple zeros of r(z) are the positions of equilibrium in the field of force due to particles of positive mass at the zeros of r(z) and particles of
negative mass In theoretical physics, negative mass is a hypothetical type of exotic matter whose mass is of opposite sign to the mass of normal matter, e.g. −1 kg. Such matter would violate one or more energy conditions and exhibit strange properties ...
at the
poles Pole or poles may refer to: People *Poles (people), another term for Polish people, from the country of Poland * Pole (surname), including a list of people with the name * Pole (musician) (Stefan Betke, born 1967), German electronic music artist ...
of r(z), with masses numerically equal to the respective multiplicities, where each particle repels with a force equal to the mass times the inverse distance.


Bôcher's equation

Bôcher's equation is a second-order
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematic ...
of the form: :y'' + \frac \left frac+ \cdots +\frac \righty' + \frac \left frac \right = 0.


The Bôcher Memorial Prize

The
Bôcher Memorial Prize The Bôcher Memorial Prize was founded by the American Mathematical Society in 1923 in memory of Maxime Bôcher with an initial endowment of $1,450 (contributed by members of that society). It is awarded every three years (formerly every five yea ...
is awarded by the American Mathematical Society every five years for notable research in
analysis Analysis (: analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...
that has appeared in a recognized North American journal. Winners have included James W. Alexander II (1928), Eric Temple Bell (1924), George D. Birkhoff (1923), Paul J. Cohen (1964),
Solomon Lefschetz Solomon Lefschetz (; 3 September 1884 – 5 October 1972) was a Russian-born American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear ordinary differential equatio ...
(1924), Marston Morse and
Norbert Wiener Norbert Wiener (November 26, 1894 – March 18, 1964) was an American computer scientist, mathematician, and philosopher. He became a professor of mathematics at the Massachusetts Institute of Technology ( MIT). A child prodigy, Wiener late ...
(1933), and
John von Neumann John von Neumann ( ; ; December 28, 1903 – February 8, 1957) was a Hungarian and American mathematician, physicist, computer scientist and engineer. Von Neumann had perhaps the widest coverage of any mathematician of his time, in ...
(1938).


Works

* 1894
Ueber die Reihenentwicklungen der Potentialtheorie
via
Internet Archive The Internet Archive is an American 501(c)(3) organization, non-profit organization founded in 1996 by Brewster Kahle that runs a digital library website, archive.org. It provides free access to collections of digitized media including web ...
* 1900: "Randwertaufgaben bei Gewöhnlich Differentialgleichung", ''Encyclopädie der mathematischen Wissenschaften'' Band 2–1–1. * 1907: (with E.P.R.DuVal
Introduction to Higher Algebra
via
HathiTrust HathiTrust Digital Library is a large-scale collaborative repository of digital content from research libraries. Its holdings include content digitized via Google Books and the Internet Archive digitization initiatives, as well as content digit ...
* 1909
Introduction to the study of Integral Equations
via Internet Archive * 1917
Leçons sur les méthodes de Sturm dans la théorie des équations différentielles linéaires et leurs développements modernes
via Internet Archive. Bôcher was one of the editors of the ''
Annals of Mathematics The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as t ...
'' and of the ''
Transactions of the American Mathematical Society The ''Transactions of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of pure and applied mathematics published by the American Mathematical Society. It was established in 1900. As a requirement, all articles must ...
''.


References


External links

*
Maxime Bocher
biographical memoirs of the national academy of sciences. *
National Academy of Sciences Biographical Memoir
{{DEFAULTSORT:Bocher, Maxime 19th-century American mathematicians 20th-century American mathematicians American mathematical analysts Harvard University Department of Mathematics faculty Harvard University alumni University of Göttingen alumni 1867 births 1918 deaths Presidents of the American Mathematical Society Cambridge Rindge and Latin School alumni Mathematicians from Boston Members of the United States National Academy of Sciences Members of the American Philosophical Society