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Sokhotski
Julian Karol Sochocki (; ; February 2, 1842, in Warsaw, Congress Poland, Russian Empire – December 14, 1927, in Saint Petersburg, Leningrad, Soviet Union) was a Polish people, Polish-Russian mathematician. His name is sometimes transliterated from Russian in several different ways (e.g. Sokhotski or Sochotski). Life and work Sochocki was born in Warsaw under the Russian domination to a Polish family, where he attended state gymnasium. In 1860 he registered at the physico-mathematical department of Saint Petersburg State University, St Petersburg University. His study there was interrupted for the period 1860–1865 because of his involvement with Polish patriotic movement: he had to return to Warsaw to escape prosecution. In 1866 he graduated from the Department of Physics and Mathematics at the Saint Petersburg State University, University of Saint Petersburg. In 1868 he received his master's degree and in 1873 his Doctor of Philosophy, doctorate. His master's dissertation, p ...
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Sokhotski–Plemelj Theorem
The Sokhotski–Plemelj theorem (Polish spelling is ''Sochocki'') is a theorem in complex analysis, which helps in evaluating certain integrals. The real-line version of it (#Version for the real line, see below) is often used in physics, although rarely referred to by name. The theorem is named after Julian Sochocki, who proved it in 1868, and Josip Plemelj, who rediscovered it as a main ingredient of his solution of the Riemann–Hilbert problem in 1908. Statement of the theorem Let ''C'' be a smooth Jordan curve, closed simple curve in the plane, and \varphi an Holomorphic function, analytic function on ''C''. Note that the Cauchy's integral formula, Cauchy-type integral : \phi(z) = \frac \int_C\frac, cannot be evaluated for any ''z'' on the curve ''C''. However, on the interior and exterior of the curve, the integral produces analytic functions, which will be denoted \phi_i inside ''C'' and \phi_e outside. The Sokhotski–Plemelj formulas relate the limiting boundary v ...
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Sokhotski–Plemelj Theorem
The Sokhotski–Plemelj theorem (Polish spelling is ''Sochocki'') is a theorem in complex analysis, which helps in evaluating certain integrals. The real-line version of it ( see below) is often used in physics, although rarely referred to by name. The theorem is named after Julian Sochocki, who proved it in 1868, and Josip Plemelj, who rediscovered it as a main ingredient of his solution of the Riemann–Hilbert problem in 1908. Statement of the theorem Let ''C'' be a smooth closed simple curve in the plane, and \varphi an analytic function on ''C''. Note that the Cauchy-type integral : \phi(z) = \frac \int_C\frac, cannot be evaluated for any ''z'' on the curve ''C''. However, on the interior and exterior of the curve, the integral produces analytic functions, which will be denoted \phi_i inside ''C'' and \phi_e outside. The Sokhotski–Plemelj formulas relate the limiting boundary values of these two analytic functions at a point ''z'' on ''C'' and the Cauchy princip ...
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Casorati–Sokhotski–Weierstrass Theorem
In complex analysis, a branch of mathematics, the Casorati–Weierstrass theorem describes the behaviour of holomorphic functions near their essential singularities. It is named for Karl Theodor Wilhelm Weierstrass and Felice Casorati. In Russian literature it is called Sokhotski's theorem, because it was discovered independently by Sokhotski in 1868. Formal statement of the theorem Start with some open subset U in the complex plane containing the number z_0, and a function f that is holomorphic on U \setminus \, but has an essential singularity at z_0 . The ''Casorati–Weierstrass theorem'' then states that This can also be stated as follows: Or in still more descriptive terms: The theorem is considerably strengthened by Picard's great theorem, which states, in the notation above, that f assumes ''every'' complex value, with one possible exception, infinitely often on V. In the case that f is an entire function and a = \infty, the theorem says that the values f(z) ...
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Warsaw
Warsaw, officially the Capital City of Warsaw, is the capital and List of cities and towns in Poland, largest city of Poland. The metropolis stands on the Vistula, River Vistula in east-central Poland. Its population is officially estimated at 1.86 million residents within a Warsaw metropolitan area, greater metropolitan area of 3.27 million residents, which makes Warsaw the List of cities in the European Union by population within city limits, 6th most-populous city in the European Union. The city area measures and comprises List of districts and neighbourhoods of Warsaw, 18 districts, while the metropolitan area covers . Warsaw is classified as an Globalization and World Cities Research Network#Alpha 2, alpha global city, a major political, economic and cultural hub, and the country's seat of government. It is also the capital of the Masovian Voivodeship. Warsaw traces its origins to a small fishing town in Masovia. The city rose to prominence in the late 16th cent ...
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Residue Theorem
In complex analysis, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as well. It generalizes the Cauchy integral theorem and Cauchy's integral formula. The residue theorem should not be confused with special cases of the generalized Stokes' theorem; however, the latter can be used as an ingredient of its proof. Statement of Cauchy's residue theorem The statement is as follows: Residue theorem: Let U be a simply connected open subset of the complex plane containing a finite list of points a_1, \ldots, a_n, U_0 = U \smallsetminus \, and a function f holomorphic function, holomorphic on U_0. Letting \gamma be a closed rectifiable curve in U_0, and denoting the residue (complex analysis), residue of f at each point a_k by \operatorname(f, a_k) and the winding number of \gamma around a_k by \operatorname(\gamma, a ...
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19th-century Mathematicians From The Russian Empire
The 19th century began on 1 January 1801 (represented by the Roman numerals MDCCCI), and ended on 31 December 1900 (MCM). It was the 9th century of the 2nd millennium. It was characterized by vast social upheaval. Slavery was abolished in much of Europe and the Americas. The First Industrial Revolution, though it began in the late 18th century, expanded beyond its British homeland for the first time during the 19th century, particularly remaking the economies and societies of the Low Countries, France, the Rhineland, Northern Italy, and the Northeastern United States. A few decades later, the Second Industrial Revolution led to ever more massive urbanization and much higher levels of productivity, profit, and prosperity, a pattern that continued into the 20th century. The Catholic Church, in response to the growing influence and power of modernism, secularism and materialism, formed the First Vatican Council in the late 19th century to deal with such problems and confirm cer ...
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Polish Mathematicians
Polish may refer to: * Anything from or related to Poland, a country in Europe * Polish language * Polish people, people from Poland or of Polish descent * Polish chicken * Polish brothers (Mark Polish and Michael Polish, born 1970), American twin screenwriters * Kevin Polish, an American Paralympian archer Polish may refer to: * Polishing, the process of creating a smooth and shiny surface by rubbing or chemical action ** French polishing, polishing wood to a high gloss finish * Nail polish * Shoe polish * Polish (screenwriting), improving a script in smaller ways than in a rewrite See also * * * Polishchuk (surname) * Polonaise (other) A polonaise ()) is a stately dance of Polish origin or a piece of music for this dance. Polonaise may also refer to: * Polonaises (Chopin), compositions by Frédéric Chopin ** Polonaise in A-flat major, Op. 53 (, ''Heroic Polonaise''; ) * Polon ... {{Disambiguation, surname Language and nationality disambiguation pages ...
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Scientists From Warsaw
A scientist is a person who researches to advance knowledge in an area of the natural sciences. In classical antiquity, there was no real ancient analog of a modern scientist. Instead, philosophers engaged in the philosophical study of nature called natural philosophy, a precursor of natural science. Though Thales ( 624–545 BC) was arguably the first scientist for describing how cosmic events may be seen as natural, not necessarily caused by gods,Frank N. Magill''The Ancient World: Dictionary of World Biography'', Volume 1 Routledge, 2003 it was not until the 19th century that the term ''scientist'' came into regular use after it was coined by the theologian, philosopher, and historian of science William Whewell in 1833. History The roles of "scientists", and their predecessors before the emergence of modern scientific disciplines, have evolved considerably over time. Scientists of different eras (and before them, natural philosophers, mathematicians, natur ...
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Russian People Of Polish Descent
Russian(s) may refer to: *Russians (), an ethnic group of the East Slavic peoples, primarily living in Russia and neighboring countries *A citizen of Russia *Russian language, the most widely spoken of the Slavic languages *''The Russians'', a book by Hedrick Smith *Russian (comics), fictional Marvel Comics supervillain from ''The Punisher'' series *Russian (solitaire), a card game * "Russians" (song), from the album ''The Dream of the Blue Turtles'' by Sting *"Russian", from the album ''Tubular Bells 2003'' by Mike Oldfield *"Russian", from the album '' '' by Caravan Palace *Nik Russian, the perpetrator of a con committed in 2002 See also * *Russia (other) *Rus (other) *Rossiysky (other) Rossiysky (masculine), Rossiyskaya (feminine), or Rossiyskoye (neuter), all meaning ''Russian Federation, Russian'', may refer to: *Rossiysky, Orenburg Oblast, a rural locality (a settlement) in Orenburg Oblast, Russia *Rossiysky, Rostov Oblast, a r ... * Russian River ...
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Saint Petersburg State University Alumni
In Christian belief, a saint is a person who is recognized as having an exceptional degree of holiness, likeness, or closeness to God. However, the use of the term ''saint'' depends on the context and denomination. In Anglican, Oriental Orthodox, and Lutheran doctrine, all of their faithful deceased in Heaven are considered to be saints, but a selected few are considered worthy of greater honor or emulation. Official ecclesiastical recognition, and veneration, is conferred on some denominational saints through the process of canonization in the Catholic Church or glorification in the Eastern Orthodox Church after their approval. In many Protestant denominations, and following from Pauline usage, ''saint'' refers broadly to any holy Christian, without special recognition or selection. While the English word ''saint'' (deriving from the Latin ) originated in Christianity, historians of religion tend to use the appellation "in a more general way to refer to the state of special ...
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Number Theorists
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations ( Diophantine geometry). Questions in number theory can often be understood through the study of analytical objects, such as the Riemann zeta function, that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, as for instance how irrational numbers can be approximated by fractions (Diophantine approximation). Number theory is one of the oldest branches of mathematics alongside geometry. One quirk of number theory is that it deals wi ...
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