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 Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...

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 Silesia, Czech Republic and Germany. Its area is approximately , and the population is estimated at around 8,000,000. S ...

(now Jerzmanowa,

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 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 Chemistry is the scientific study of the properties and behavior of matter. It is a natural science that covers the elements that make up matter to the compounds made of atoms, molecules and ions: their composition, structure, proper ...

in Berlin but Ernst Eduard Kummer and

Karl Theodor Wilhelm Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...

persuaded him to change to

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 ori ...

, dealing with the subjects of

differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...

and the

calculus of variations The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...

. He died in

Berlin Berlin ( , ) is the capital and largest city of Germany by both area and population. Its 3.7 million inhabitants make it the European Union's most populous city, according to population within city limits. One of Germany's sixteen constitu ...

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 The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is considered one of the most important and widely used inequalities in mathematics. The inequality for sums was published by . The corresponding inequality f ...

, 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 ...

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 Unite ...

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, Paul Koebe 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 ...

. 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 * Additive Schwarz method * Schwarz alternating method * * Schwarzian derivative * Schwarz function *

Schwarz lantern In mathematics, the Schwarz lantern is a polyhedron, polyhedral approximation to a cylinder, used as a Pathological (mathematics), pathological example of the difficulty of defining the area of a smooth (curved) surface as the Limit (mathemati ...

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 * Schwarz theorem (also known as Clairaut's theorem) * Schwarz integral formula * Schwarz–Christoffel mapping *

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 *

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 define ...

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 necessa ...

* 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