Phrase guide

Möbius transformation

noun · 1 senses · updated from the 2026-07-25 local source snapshot

Definitions and examples are grouped by meaning. Pronunciation, history, word forms, translations, descendants, synonyms, antonyms, derived terms, and related words appear whenever the source provides them.

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Pronunciation

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1

noun

Meaning 1

A transformation of the extended complex plane that is a rational function of the form f(z) = (az + b) / (cz + d), where a, b, c, d are complex numbers such that ad − bc ≠ 0; an automorphism of the complex projective line.

Definition source: English Wiktionary via Wiktextract

Topics: complex-analysis, geometry, mathematics, sciences

Examples

  • 2008, J. Vernon Armitage, John Parker, Jørgensen's inequality for Non-Archimedean Metric Spaces, Mikhail Kapranov, Sergii Kolyada, Yu. I. Manin, Pieter Moree, Leonid Potyagailo (editors), Geometry and Dynamics of Groups and Spaces: In Memory of Alexander Reznikov, page 97, Jørgensen's inequality gives a necessary condition for a non-elementary group of Möbius transformations to be discrete.
  • 2012, Martin Delacourt, Petr Kůrka, Finite State Transducers of Modular Möbius Number Systems, Branislav Rovan, Vladimiro Sassone, Peter Widmayer (editors), Mathematical Foundations of Computer Science 2012, 37th International Symposium, MFCS 2012, Proceedings, Springer, LNCS 7464, page 323, Modular Möbius number systems consist of Möbius transformations with integer coefficients and unit determinant.

Meaning relationships

Synonyms: none provided

Antonyms: none provided

History

Etymology

Named for German mathematician and theoretical astronomer August Ferdinand Möbius (1790–1868).

Across languages

Translations

7 source translations are retained for this English entry.

  • Dutch: Möbius-transformatie — transformation of the complex plane
  • French: transformation de Möbius — transformation of the complex plane
  • German: Möbiustransformation — transformation of the complex plane
  • Italian: trasformazione di Möbius — transformation of the complex plane
  • Polish: funkcja homograficzna — transformation of the complex plane
  • Portuguese: transformação de Möbius — transformation of the complex plane
  • Spanish: transformación de Möbius — transformation of the complex plane