{{Infobox scientist , name = Lucjan Emil Böttcher , native_name = , native_name_lang = , image = , image_size = , alt = , caption = , birth_date = , birth_place =

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, nationality = Polish , fields = , workplaces = , alma_mater = Lwów Polytechnic School,

University of Leipzig Leipzig University (german: Universität Leipzig), in Leipzig in Saxony, Germany, is one of the world's oldest universities and the second-oldest university (by consecutive years of existence) in Germany. The university was founded on 2 Decemb ...

, thesis_title = Beiträge zu der Theorie der Iterationsrechnung , thesis_url = , thesis_year = 1898 , doctoral_advisor =

Sophus Lie Marius Sophus Lie ( ; ; 17 December 1842 – 18 February 1899) was a Norwegian mathematician. He largely created the theory of continuous symmetry and applied it to the study of geometry and differential equations. Life and career Marius Sophu ...

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Böttcher's equation Böttcher's equation, named after Lucjan Böttcher, is the functional equation ::F(h(z)) = (F(z))^n where * is a given analytic function with a superattracting fixed point of order at , (that is, h(z)=a+c(z-a)^n+O((z-a)^) ~, in a neighbour ...

, author_abbrev_bot = , author_abbrev_zoo = , influences = , influenced = , awards = , signature = , signature_alt = , website = , footnotes = , spouse = , children = Lucjan Emil Böttcher (January 21, 1872 – May 29, 1937) was a Polish

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

who worked in Lvov in the beginning of the 20th century.{{cite arXiv , first=Stanisław , last=Domoradzki , first2=Małgorzata , last2=Stawiska , date= July 11, 2012 , title=Lucjan Emil Böttcher and his mathematical legacy , class=math.HO , eprint=1207.2747

Early life

Böttcher was born on January 21, 1872, in

. He attended private schools in Warsaw and graduated from the classical gymnasium in

Łomża Łomża (), in English known as Lomza, is a city in north-eastern Poland, approximately 150 kilometers (90 miles) to the north-east of Warsaw and west of Białystok. It is situated alongside the Narew river as part of the Podlaskie Voivodeship si ...

in 1893, after which he entered the

Imperial University of Warsaw The University of Warsaw ( pl, Uniwersytet Warszawski, la, Universitas Varsoviensis) is a public university in Warsaw, Poland. Established in 1816, it is the largest institution of higher learning in the country offering 37 different fields of ...

in the Division of Mathematics and Physics. At the time, Russian was the language of instruction at the university, as Warsaw was under Russian rule. He was expelled from the university for participation in patriotic (anti-Russian) demonstrations in 1894. He moved to Lwów Polytechnic School, where he obtained a so-called half-diploma in 1897. Desiring to continue his mathematical education, he moved to

Leipzig Leipzig ( , ; Upper Saxon: ) is the most populous city in the German state of Saxony. Leipzig's population of 605,407 inhabitants (1.1 million in the larger urban zone) as of 2021 places the city as Germany's eighth most populous, as wel ...

, where he worked under

. His doctoral thesis, published in 1898, was titled ''Beiträge zu der Theorie der Iterationsrechnung''. Böttcher married Maria Wolle in 1900, and had four children.

Career

Following his doctorate, Böttcher returned to Lwów to take up a junior position at the Lwów Polytechnic School. By 1911, he was licensed to teach (''venia legendi'') at the school, and he offered courses on theoretical mechanics as well as mathematics for engineering. All his attempts to obtain

habilitation Habilitation is the highest university degree, or the procedure by which it is achieved, in many European countries. The candidate fulfills a university's set criteria of excellence in research, teaching and further education, usually including a ...

at the University of Lwów failed, however. This meant that he was not permitted to guide doctoral students.{{cite arXiv , last=Stawiska, first= Małgorzata, author-link=Małgorzata Stawiska, eprint=1307.7778 , title=Lucjan Emil Böttcher (1872–1937) - The Polish Pioneer of Holomorphic Dynamics, class= math.HO, date=November 15, 2013 Böttcher was a member of the Polish Mathematical Society. He took seriously his role of an educator, encouraging the introduction of differential and integral calculus at school level, and writing several high-school textbooks. One example is ''Principles of Elementary Algebra, adapted to the curriculum in the Polish Kingdom'' (1911), which followed the so-called ''Meran programme'' that aimed to teach students to think in terms of functions.

Main works

Böttcher's most important work was in the iterations of

rational mapping In mathematics, in particular the subfield of algebraic geometry, a rational map or rational mapping is a kind of partial function between algebraic varieties. This article uses the convention that varieties are irreducible. Definition Formal de ...

s of the

Riemann sphere In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers pl ...

. His name is attached to Böttcher theorem, in which he introduced

and solved it under certain assumptions. He obtained results about the orbits of iterated rational maps, studied their convergence regions ('' Fatou components'') and boundaries ('' Julia set''); he also gave examples of everywhere chaotic maps constructed via

elliptic function In the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those in ...

s. Indeed, his example of rational maps whose chaotic set is the entire sphere predated the more famous ''Lattès examples'' by over twenty years.{{cite journal, journal=Leningrad Math. J. , volume=1, year=1990, number=3, pages=563–634, title=The dynamics of analytic transformations, first1=A.E., last1=Eremenko, first2=M.Yu., last2=Lyubich

Academic reception

Böttcher was one of the founders of

Holomorphic dynamics Complex dynamics is the study of dynamical systems defined by iteration of functions on complex number spaces. Complex analytic dynamics is the study of the dynamics of specifically analytic functions. Techniques *General **Montel's theorem **Po ...

, which he viewed as a part of the mathematical theory of iterational calculus. Despite his achievements, however, his early publications were considered insufficient to warrant a habilitation at the University of Lwow in 1901. Seventeen years later, with more publications to his name, he approached the University again for habilitation, but his request was denied. The committee's criticism focused on erroneous and unscientific reasoning in some of his papers, and cited the lack of clarity even in his expository works. As Böttcher worked in a mathematical discipline considerably removed from the interests of other Lwów mathematicians, he found little support from his peers. Indeed, his partial results and conclusions were forgotten, and a complete theory came about only decades later following the independent investigations of

Pierre Fatou Pierre Joseph Louis Fatou (28 February 1878 – 9 August 1929) was a French mathematician and astronomer. He is known for major contributions to several branches of analysis. The Fatou lemma and the Fatou set are named after him. Biography P ...

Gaston Julia Gaston Maurice Julia (3 February 1893 – 19 March 1978) was a French Algerian mathematician who devised the formula for the Julia set. His works were popularized by French mathematician Benoit Mandelbrot; the Julia and Mandelbrot fractals are cl ...

Samuel Lattès Samuel Lattès (21 February 1873 (Nice) – 5 July 1918) was a French mathematician. From 1892 to 1895 he studied at the École Normale Superieure. After this he was a teacher in Algiers, Dijon and Nice. After a promotion to Paris in 1906 he moved ...

and Salvatore Pincherle.

Later life

Böttcher retired from the Polytechnic School in 1935. He died in Lwów on May 29, 1937.

Selected bibliography

Papers

# {{Cite thesis , title=Beiträge zu der Theorie der Iterationsrechnung , type= Ph.D., year= 1898, publisher=Oswald Schmidt # {{cite journal, title=Zasady rachunku iteracyjnego (część pierwsza, część druga), journal= Prace Matematyczno Fizyczne, volume= X, year=1899–1900, pages=65-86, 86-101 # {{cite journal, title=Zasady rachunku iteracyjnego (część III), journal=Prace Matematyczno Fizyczne, volume=XII, year=1901, pages=95–111 # {{cite journal, title=Zasady rachunku iteracyjnego (część III, dokończenie), journal=Prace Matematyczno Fizyczne, volume=XIII, year=1902, pages=353–371 # {{cite journal, title=Glavnyshiye zakony skhodimosti iteratsiy i ikh prilozheniya k' analizu, journal=Bulletin de la Société Physico-Mathématique de Kasan, volume=XIII, number=1, year=1903, page=137 # {{cite journal, title=Glavnyshiye zakony skhodimosti iteratsiy i ikh prilozheniya k' analizu, journal=Bulletin de la Société Physico-Mathématique de Kasan, volume=XIV, number=2, year=1904, pages=155–200 # {{cite journal, title=Glavnyshiye zakony skhodimosti iteratsiy i ikh prilozheniya k' analizu, journal=Bulletin de la Société Physico-Mathématique de Kasan, volume=XIV, number=3, year=1904, pages=201–234

Textbooks

# {{cite book, title=Principles of geometry with numerous exercises, year=1908, location=Warsaw, publisher=M. Arct # {{cite book, title=Principles of Elementary Algebra, adapted to the curriculum in the Polish Kingdom, year=1911, location=Warsaw

References

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