# Hermann Schwarz Information (Person)

*https://en.wikipedia.org/wiki/Hermann_Schwarz*

Hermann Schwarz | |
---|---|

Born | |

Died | 30 November 1921 | (aged 78)

Nationality | Prussian |

Alma mater | Gewerbeinstitut |

Known for | Cauchy–Schwarz inequality |

Scientific career | |

Fields | Mathematician |

Institutions |
University of Halle Swiss Federal Polytechnic Göttingen University |

Doctoral advisor |
Karl Weierstrass Ernst Kummer |

Doctoral students |
Lipót 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.

## Life

Schwarz was born in
Hermsdorf,
Silesia (now
Jerzmanowa,
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.

## 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,^{
[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:

- Abstract additive Schwarz method
- Additive Schwarz method
- Schwarz alternating method
- Schwarzian derivative
- Schwarz function
- Schwarz lantern
- Schwarz lemma
- Schwarz's list
- 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
- Schwarz triangle function
- Cauchy–Schwarz inequality
- Theorem of Pohlke and Schwarz

## Publications

- 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

## Notes

**^**Carathéodory, C (1921). "Hermann Amandus Schwarz".*Deutsches biographisches Jahrbuch*.**III**: 236-238. Retrieved 7 July 2021.- ^
^{a}^{b}^{c}Agarwal, Ravi; Sen, Syamal (11 November 2014).*Creators of Mathematical and Computational Sciences*. Springer. pp. 297–298. ISBN 9783319108704. - ^
^{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. - ^
^{a}^{b}"The Mathematics Genealogy Project – Hermann Schwarz".*www.genealogy.math.ndsu.nodak.edu*. Retrieved 22 May 2016. - ^
^{a}^{b}Chang, Sooyoung (1 January 2011).*Academic Genealogy of Mathematicians*. World Scientific. pp. 77–78. ISBN 9789814282291. **^**Bottazzini, Umberto (30 April 2003). "Algebraic truths vs geometric fantasies: Weierstrass' Response to Riemann". arXiv: math/0305022.**^**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.**^**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.

## External links

- 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