Karl Hermann Amandus Schwarz (; 25 January 1843 – 30 November 1921) 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 ...

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 his work in complex analysis.

Life

Schwarz was born in Hermsdorf,

Silesia Silesia (, also , ) is a historical region of Central Europe that lies mostly within Poland, with small parts in the Czech Republic and Germany. Its area is approximately , and the population is estimated at around 8,000,000. Silesia is split ...

(now

Jerzmanowa Jerzmanowa (; german: Hermsdorf, ) is a village in Głogów County, Lower Silesian Voivodeship, in south-western Poland. It is the seat of the administrative district ( gmina) called Gmina Jerzmanowa. The village has an approximate population ...

Poland Poland, officially the Republic of Poland, is a country in Central Europe. It is divided into 16 administrative provinces called voivodeships, covering an area of . Poland has a population of over 38 million and is the fifth-most populou ...

). In 1868 he married Marie Kummer, who was the daughter to the mathematician

Ernst Eduard Kummer Ernst Eduard Kummer (29 January 1810 – 14 May 1893) was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a '' gymnasium'', the German equivalent of h ...

and Ottilie née Mendelssohn (a daughter of Nathan Mendelssohn's and granddaughter of

Moses Mendelssohn Moses Mendelssohn (6 September 1729 – 4 January 1786) was a German-Jewish philosopher and theologian. His writings and ideas on Jews and the Jewish religion and identity were a central element in the development of the ''Haskalah'', or ' ...

). Schwarz and Kummer had six children, including his daughter Emily Schwarz. Schwarz originally studied chemistry in Berlin but Ernst Eduard Kummer and Karl Theodor Wilhelm Weierstrass persuaded him to change to mathematics. He received his Ph.D. from the Universität Berlin in 1864 and was advised by Kummer and Weierstrass. Between 1867 and 1869 he worked at the

University of Halle Martin Luther University of Halle-Wittenberg (german: Martin-Luther-Universität Halle-Wittenberg), also referred to as MLU, is a public, research-oriented university in the cities of Halle and Wittenberg and the largest and oldest university in ...

, then at the Swiss Federal Polytechnic. From 1875 he worked at

Göttingen University Göttingen (, , ; nds, Chöttingen) is a university city in Lower Saxony, central Germany, the capital of the eponymous district. The River Leine runs through it. At the end of 2019, the population was 118,911. General information The o ...

, dealing with the subjects of complex analysis, differential geometry and the calculus of variations. He died in

Berlin Berlin ( , ) is the capital and List of cities in Germany by population, largest city of Germany by both area and population. Its 3.7 million inhabitants make it the European Union's List of cities in the European Union by population within ci ...

Work

Schwarz's works include ''Bestimmung einer speziellen Minimalfläche'', which was crowned by the Berlin Academy in 1867 and printed in 1871, and ''Gesammelte mathematische Abhandlungen'' (1890). Among other things, Schwarz improved the proof of the

Riemann mapping theorem In complex analysis, the Riemann mapping theorem states that if ''U'' is a non-empty simply connected open subset of the complex number plane C which is not all of C, then there exists a biholomorphic mapping ''f'' (i.e. a bijective holomorphi ...

, developed a special case of the Cauchy–Schwarz inequality, and gave a proof that the ball has less surface area than any other body of equal volume. His work on the latter allowed

Émile Picard Charles Émile Picard (; 24 July 1856 – 11 December 1941) was a French mathematician. He was elected the fifteenth member to occupy seat 1 of the Académie française in 1924. Life He was born in Paris on 24 July 1856 and educated there at t ...

to show solutions of differential equations exist (the

Picard–Lindelöf theorem In mathematics – specifically, in differential equations – the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. It is also known as Picard's existence theorem, the Cau ...

). In 1892 he became a member of the

Academy of Science An academy of sciences is a type of learned society or academy (as special scientific institution) dedicated to sciences that may or may not be state funded. Some state funded academies are tuned into national or royal (in case of the Unit ...

and a professor at the

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 ...

, where his students included

Lipót Fejér Lipót Fejér (or Leopold Fejér, ; 9 February 1880 – 15 October 1959) was a Hungarian mathematician of Jewish heritage. Fejér was born Leopold Weisz, and changed to the Hungarian name Fejér around 1900. Biography Fejér studied mathematic ...

Paul Koebe Paul Koebe (15 February 1882 – 6 August 1945) was a 20th-century German mathematician. His work dealt exclusively with the complex numbers, his most important results being on the uniformization of Riemann surfaces in a series of four papers in ...

and

Ernst Zermelo Ernst Friedrich Ferdinand Zermelo (, ; 27 July 187121 May 1953) was a German logician and mathematician, whose work has major implications for the foundations of mathematics. He is known for his role in developing Zermelo–Fraenkel axiomatic se ...

. In total, he advised at least 22 Ph. D students. In 1914 Schwarz's friends and former students published a volume with 34 articles in celebration of the 50th anniversary of his doctoral dissertation. His name is attached to many ideas in mathematics, including the following: *

Abstract additive Schwarz method Abstract may refer to: * ''Abstract'' (album), 1962 album by Joe Harriott * Abstract of title a summary of the documents affecting title to parcel of land * Abstract (law), a summary of a legal document * Abstract (summary), in academic publish ...

Additive Schwarz method In mathematics, the additive Schwarz method, named after Hermann Schwarz, solves a boundary value problem for a partial differential equation approximately by splitting it into boundary value problems on smaller domains and adding the results. ...

Schwarz alternating method Schwarz may refer to: * Schwarz, Germany, a municipality in Mecklenburg-Vorpommern, Germany * Schwarz (surname), a surname (and list of people with the surname) * Schwarz (musician), American DJ and producer * Schwarz (Böhse Onkelz album), ''Schw ...

* *

Schwarzian derivative In mathematics, the Schwarzian derivative is an operator similar to the derivative which is invariant under Möbius transformations. Thus, it occurs in the theory of the complex projective line, and in particular, in the theory of modular forms an ...

* Schwarz function * Schwarz lantern *

Schwarz lemma In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions from the open unit disk to itself. The lemma is less celebrated than deeper theorems, such as the Riemann mapp ...

* Schwarz's list *

Schwarz minimal surface In differential geometry, the Schwarz minimal surfaces are periodic minimal surfaces originally described by Hermann Schwarz. In the 1880s Schwarz and his student E. R. Neovius described periodic minimal surfaces. They were later named ...

Schwarz theorem In mathematics, the symmetry of second derivatives (also called the equality of mixed partials) refers to the possibility of interchanging the order of taking partial derivatives of a function :f\left(x_1,\, x_2,\, \ldots,\, x_n\right) of ''n ...

(also known as Clairaut's theorem) *

Schwarz integral formula In complex analysis, a branch of mathematics, the Schwarz integral formula, named after Hermann Schwarz, allows one to recover a holomorphic function, up to an imaginary constant, from the boundary values of its real part. Unit disc Let ''f'' be a ...

Schwarz–Christoffel mapping In complex analysis, a Schwarz–Christoffel mapping is a conformal map of the upper half-plane or the complex unit disk onto the interior of a simple polygon. Such a map is guaranteed to exist by the Riemann mapping theorem (stated by Bernhard ...

Schwarz–Ahlfors–Pick theorem In mathematics, the Schwarz–Ahlfors–Pick theorem is an extension of the Schwarz lemma for hyperbolic geometry, such as the Poincaré half-plane model. The Schwarz–Pick lemma states that every holomorphic function from the unit disk ''U' ...

Schwarz reflection principle In mathematics, the Schwarz reflection principle is a way to extend the domain of definition of a complex analytic function, i.e., it is a form of analytic continuation. It states that if an analytic function is defined on the upper half-plane, a ...

Schwarz triangle In geometry, a Schwarz triangle, named after Hermann Schwarz, is a spherical triangle that can be used to tile a sphere (spherical tiling), possibly overlapping, through reflections in its edges. They were classified in . These can be defined mor ...

Schwarz triangle function In complex analysis, the Schwarz triangle function or Schwarz s-function is a function that conformally maps the upper half plane to a triangle in the upper half plane having lines or circular arcs for edges. The target triangle is not necess ...

* Cauchy–Schwarz inequality * Theorem of Pohlke and Schwarz

Publications

* *

Notes

External links

* * {{DEFAULTSORT:Schwarz, Hermann 1843 births 1921 deaths 20th-century German mathematicians 19th-century German mathematicians ETH Zurich faculty University of Göttingen faculty Martin Luther University of Halle-Wittenberg faculty Humboldt University of Berlin faculty Corresponding members of the Saint Petersburg Academy of Sciences People from the Province of Silesia People from Głogów County Members of the Royal Society of Sciences in Uppsala