Hermann Schwarz

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Hermann Schwarz
Schwarz c. 1890
Born(1843-01-25)25 January 1843
Died30 November 1921(1921-11-30) (aged 78)
Nationality (legal)Prussian
Alma materGewerbeinstitut
Known forCauchy–Schwarz inequality
Scientific career
InstitutionsUniversity of Halle
Swiss Federal Polytechnic
Göttingen University
Doctoral advisorKarl Weierstrass
Ernst Kummer
Doctoral studentsLipót Fejér
Harris Hancock
Gerhard Hessenberg
Paul Koebe
Leon Lichtenstein
Heinrich Maschke
Robert Remak
Rudolf Rothe
Theodor Vahlen
Ernst Zermelo

Karl Hermann Amandus Schwarz (German: [ˈhɛʁman ˈʃvaʁts]; 25 January 1843 – 30 November 1921) was a German mathematician, known for his work in complex analysis.


Schwarz was born in Hermsdorf, Silesia (now Sobieszów, Poland). In 1868 he married Marie Kummer,[1] who was the daughter to the mathematician Ernst Eduard Kummer[2] 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.[2]

Schwarz originally studied chemistry in Berlin but Ernst Eduard Kummer and Karl Theodor Wilhelm Weierstrass persuaded him to change to mathematics.[3] He received his Ph.D. from the Universität Berlin in 1864 and was advised by Kummer and Weierstrass.[4] Between 1867 and 1869 he worked at the University of Halle, then at the Swiss Federal Polytechnic.[5] From 1875 he worked at Göttingen University,[5] dealing with the subjects of complex analysis, differential geometry and the calculus of variations. He died in Berlin.


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,[6] 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.[7] His work on the latter allowed Émile Picard to show solutions of differential equations exist (the Picard–Lindelöf theorem).[3]

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, Paul Koebe and Ernst Zermelo. In total, he advised at least 22 Ph. D students.[4] In 1914 Schwarz's friends and former students published a volume with 34 articles in celebration of the 50th anniversary of his doctoral dissertation.[8]

His name is attached to many ideas in mathematics,[2] including the following:


  • Schwarz, H. A. (1871), Bestimmung einer speziellen Minimalfläche, Dümmler
  • Schwarz, H. A. (1972) [1890], Gesammelte mathematische Abhandlungen. Band I, II, Bronx, N.Y.: AMS Chelsea Publishing, ISBN 978-0-8284-0260-6, MR 0392470


  1. ^ Carathéodory, C (1921). "Hermann Amandus Schwarz". Deutsches biographisches Jahrbuch. III (6): 236–238. Bibcode:1921ZaMM....1..494M. doi:10.1002/zamm.19210010615. Retrieved 7 July 2021.
  2. ^ a b c Agarwal, Ravi; Sen, Syamal (11 November 2014). Creators of Mathematical and Computational Sciences. Springer. pp. 297–298. ISBN 9783319108704.
  3. ^ a b O'Connor, J. J.; Robertson, E. F. "Schwarz biography". www-gap.dcs.st-and.ac.uk. The MacTutor History of Mathematics. Archived from the original on 5 June 2016. Retrieved 22 May 2016.
  4. ^ a b "The Mathematics Genealogy Project – Hermann Schwarz". www.genealogy.math.ndsu.nodak.edu. Retrieved 22 May 2016.
  5. ^ a b Chang, Sooyoung (1 January 2011). Academic Genealogy of Mathematicians. World Scientific. pp. 77–78. ISBN 9789814282291.
  6. ^ Bottazzini, Umberto (30 April 2003). "Algebraic truths vs geometric fantasies: Weierstrass' Response to Riemann". arXiv:math/0305022.
  7. ^ Schwarz, Hermann Amandus (1884). "Proof of the theorem that the ball has less surface area than any other body of the same volume". News of the Royal Society of Sciences and the Georg-August-Universität Göttingen. 1884: 1–13.
  8. ^ Gronwall, T. H. (1916). "Book Review: Mathematische Abhandlungen, Hermann Amandus Schwarz zu seinem fünfzigjährigen Doktorjubiläum am 6. August 1914 gewidmet von Freunden und Schülern". Bulletin of the American Mathematical Society. 22 (8): 406–408. doi:10.1090/S0002-9904-1916-02811-4.

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