# Hermann Schwarz

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

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

Karl Hermann Amandus Schwarz | |

Born | |

Died | 30 November 1921 78) | (aged

Residence | Germany, Switzerland |

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 Theodor Vahlen Ernst Zermelo |

## Life

Schwarz was born in Hermsdorf, Silesia (now Jerzmanowa, Poland). On 30 June 1912 he married Marie Kummer, who was the daughter to the mathematician Ernst Eduard Kummer[1] 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.[1]

Schwarz originally studied chemistry in Berlin but Ernst Eduard Kummer and Karl Theodor Wihelm Weierstrass persuaded him to change to mathematics.[2] He received his Ph.D. from the Universität Berlin in 1864 and was advised by Ernst Kummer and Karl Weierstraß.[3] Between 1867 and 1869 he worked at the University of Halle, then at the Swiss Federal Polytechnic.[4] From 1875 he worked at Göttingen University,[4] 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,[5] 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.[6] His work on the latter allowed Émile Picard to show solutions of differential equations exist (the Picard–Lindelöf theorem).[2]

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 20 Ph.D students.[3]

His name is attached to many ideas in mathematics,[1] including:

- Additive Schwarz method
- Schwarz alternating method
- Schwarzian derivative
- 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 map
- Cauchy–Schwarz inequality.

## 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

- Agarwal, Ravi; Sen, Syamal (2014-11-11).
*Creators of Mathematical and Computational Sciences*. Springer. pp. 297–298. ISBN 9783319108704. - O'Connor, J. J.; Robertson, E. F. "Schwarz biography".
*www-gap.dcs.st-and.ac.uk*. The MacTutor History of Mathematics. Retrieved 2016-05-22. - "The Mathematics Genealogy Project - Hermann Schwarz".
*www.genealogy.math.ndsu.nodak.edu*. Retrieved 2016-05-22. - Chang, Sooyoung (2011-01-01).
*Academic Genealogy of Mathematicians*. World Scientific. pp. 77–78. ISBN 9789814282291. - Bottazzini, Umberto (2003-04-30). "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.

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