Karl Hermann Amandus Schwarz (; 25 January 1843 – 30 November 1921) was a German mathematician, 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 (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 The gmina (Polish: , plural ''gminy'' , from German ''Gemeinde'' mean ...

, Poland). 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). Schwarz and Kummer had six children, including his daughter Emily Schwarz. Schwarz originally studied

chemistry Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical element, elements that make up matter to the chemical compound, compounds made of atoms, molecules and ions ...

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 mathematic ...

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, then at the

Swiss Federal Polytechnic ETH Zürich (English language, English: ETH; Swiss Federal Institute of Technology in Zürich; german: Eidgenössische Technische Hochschule Zürich) is a public university, public research university in the city of Zürich, Switzerland. Founded b ...

. From 1875 he worked at Göttingen University, 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 multili ...

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.

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, 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 fo ...

, 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 to show solutions of differential equations exist (the Picard–Lindelöf theorem). In 1892 he became a member of the Berlin Academy of Science and a professor at the University of Berlin, 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 and Ernst Zermelo. 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 The Schwarz function of a curve in the complex plane is an analytic function which maps the points of the curve to their complex conjugates. It can be used to generalize the Schwarz reflection principle to reflection across arbitrary analytic cur ...

Schwarz lantern In mathematics, the Schwarz lantern is a polyhedral approximation to a cylinder, used as a pathological example of the difficulty of defining the area of a smooth (curved) surface as the limit of the areas of polyhedra. It is formed by stack ...

* Schwarz lemma *

Schwarz's list In the mathematical theory of special functions, Schwarz's list or the Schwartz table is the list of 15 cases found by when hypergeometric functions can be expressed algebraically. More precisely, it is a listing of parameters determining the ...

* Schwarz minimal surface * Schwarz theorem (also known as Clairaut's theorem) * Schwarz integral formula * Schwarz–Christoffel mapping * Schwarz–Ahlfors–Pick theorem * 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 defin ...

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 ...

* 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