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Pochhammer
Leo August Pochhammer (25 August 1841, Stendal – 24 March 1920, Kiel) was a Prussian mathematician who was educated in Berlin, obtaining his Ph.D. in 1863 under Ernst Kummer. He became a lecturer in 1874, then professor of mathematics at the Christian-Albrechts-Universität of Kiel (1877–1919), where he was appointed Rektor (commissioner of education) in 1893. He is known for his work on special functions and introduced the Pochhammer symbol, now generally used for expressing hypergeometric functions in a compact notation. See also *Pochhammer symbol *Generalized Pochhammer symbol * ''q''-Pochhammer symbol *Pochhammer contour *Falling and rising factorials In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial :\begin (x)_n = x^\underline &= \overbrace^ \\ &= \prod_^n(x-k+1) = \prod_^(x-k) \,. \e ... External links * Eric W. Weisstein's world of science 1841 birt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pochhammer Symbol
In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial :\begin (x)_n = x^\underline &= \overbrace^ \\ &= \prod_^n(x-k+1) = \prod_^(x-k) \,. \end The rising factorial (sometimes called the Pochhammer function, Pochhammer polynomial, ascending factorial, (A reprint of the 1950 edition by Chelsea Publishing Co.) rising sequential product, or upper factorial) is defined as :\begin x^ = x^\overline &= \overbrace^ \\ &= \prod_^n(x+k-1) = \prod_^(x+k) \,. \end The value of each is taken to be 1 (an empty product) when . These symbols are collectively called factorial powers. The Pochhammer symbol, introduced by Leo August Pochhammer, is the notation , where is a non-negative integer. It may represent ''either'' the rising or the falling factorial, with different articles and authors using different conventions. Pochhammer himself actually used with yet another meaning, namely to d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Falling And Rising Factorials
In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial :\begin (x)_n = x^\underline &= \overbrace^ \\ &= \prod_^n(x-k+1) = \prod_^(x-k) \,. \end The rising factorial (sometimes called the Pochhammer function, Pochhammer polynomial, ascending factorial, (A reprint of the 1950 edition by Chelsea Publishing Co.) rising sequential product, or upper factorial) is defined as :\begin x^ = x^\overline &= \overbrace^ \\ &= \prod_^n(x+k-1) = \prod_^(x+k) \,. \end The value of each is taken to be 1 (an empty product) when . These symbols are collectively called factorial powers. The Pochhammer symbol, introduced by Leo August Pochhammer, is the notation , where is a non-negative integer. It may represent ''either'' the rising or the falling factorial, with different articles and authors using different conventions. Pochhammer himself actually used with yet another meaning, namely to d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Q-Pochhammer Symbol
In mathematical area of combinatorics, the ''q''-Pochhammer symbol, also called the ''q''-shifted factorial, is the product (a;q)_n = \prod_^ (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^), with (a;q)_0 = 1. It is a ''q''-analog of the Pochhammer symbol (x)_n = x(x+1)\dots(x+n-1), in the sense that \lim_ \frac = (x)_n. The ''q''-Pochhammer symbol is a major building block in the construction of ''q''-analogs; for instance, in the theory of basic hypergeometric series, it plays the role that the ordinary Pochhammer symbol plays in the theory of generalized hypergeometric series. Unlike the ordinary Pochhammer symbol, the ''q''-Pochhammer symbol can be extended to an infinite product: (a;q)_\infty = \prod_^ (1-aq^k). This is an analytic function of ''q'' in the interior of the unit disk, and can also be considered as a formal power series in ''q''. The special case \phi(q) = (q;q)_\infty=\prod_^\infty (1-q^k) is known as Euler's function, and is important in combinatorics, number theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pochhammer Contour
In mathematics, the Pochhammer contour, introduced by Jordan (1887), pp. 243–244 and , is a contour in the complex plane with two points removed, used for contour integration. If ''A'' and ''B'' are loops around the two points, both starting at some fixed point ''P'', then the Pochhammer contour is the commutator ''ABA''−1''B''−1, where the superscript −1 denotes a path taken in the opposite direction. With the two points taken as 0 and 1, the fixed basepoint ''P'' being on the real axis between them, an example is the path that starts at ''P'', encircles the point 1 in the counter-clockwise direction and returns to ''P'', then encircles 0 counter-clockwise and returns to ''P'', after that circling 1 and then 0 clockwise, before coming back to ''P''. The class of the contour is an actual commutator when it is considered in the fundamental group with basepoint ''P'' of the complement in the complex plane (or Riemann sphere) of the two points looped. When it come ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Pochhammer Symbol
In mathematics, the generalized Pochhammer symbol of parameter \alpha>0 and partition \kappa=(\kappa_1,\kappa_2,\ldots,\kappa_m) generalizes the classical Pochhammer symbol, named after Leo August Pochhammer, and is defined as :(a)^_\kappa=\prod_^m \prod_^ \left(a-\frac+j-1\right). It is used in multivariate analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 .... References * Gamma and related functions Factorial and binomial topics {{numtheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stendal
The Hanseatic City of Stendal () is a town in Saxony-Anhalt, Germany. It is the capital of the Stendal District and the unofficial capital of the Altmark region. Geography Situated west of the Elbe valley, the Stendal town centre is located some west of Berlin, around east of Hanover, and north of the state capital Magdeburg. Stendal is the seat of a University of Applied Sciences (''Fachhochschule'') and preserves a picturesque old town including a historic market and several churches. The nearby village Uchtspringe is home to a psychiatric rehabilitation clinic. Divisions The town Stendal consists of Stendal proper and the following 18 ''Ortschaften'' or municipal divisions:Hauptsatzung der Hansestadt Stendal November 2018. *Bindfelde *Borstel * [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
People From The Province Of Saxony
A person ( : people) is a being that has certain capacities or attributes such as reason, morality, consciousness or self-consciousness, and being a part of a culturally established form of social relations such as kinship, ownership of property, or legal responsibility. The defining features of personhood and, consequently, what makes a person count as a person, differ widely among cultures and contexts. In addition to the question of personhood, of what makes a being count as a person to begin with, there are further questions about personal identity and self: both about what makes any particular person that particular person instead of another, and about what makes a person at one time the same person as they were or will be at another time despite any intervening changes. The plural form "people" is often used to refer to an entire nation or ethnic group (as in "a people"), and this was the original meaning of the word; it subsequently acquired its use as a plural form of per ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
19th-century German Mathematicians
The 19th (nineteenth) century began on 1 January 1801 ( MDCCCI), and ended on 31 December 1900 ( MCM). The 19th century was the ninth century of the 2nd millennium. The 19th century 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, expanding beyond its British homeland for the first time during this century, particularly remaking the economies and societies of the Low Countries, 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 Islamic gunpowder empires fell into decline and European imperialism brought much of South Asia, Southeast Asia, and almost all of Africa under colonial rule. It was also marked by the collapse of the large S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1920 Deaths
Nineteen or 19 may refer to: * 19 (number), the natural number following 18 and preceding 20 * one of the years 19 BC, AD 19, 1919, 2019 Films * ''19'' (film), a 2001 Japanese film * ''Nineteen'' (film), a 1987 science fiction film Music * 19 (band), a Japanese pop music duo Albums * ''19'' (Adele album), 2008 * ''19'', a 2003 album by Alsou * ''19'', a 2006 album by Evan Yo * ''19'', a 2018 album by MHD * ''19'', one half of the double album ''63/19'' by Kool A.D. * ''Number Nineteen'', a 1971 album by American jazz pianist Mal Waldron * ''XIX'' (EP), a 2019 EP by 1the9 Songs * "19" (song), a 1985 song by British musician Paul Hardcastle. * "Nineteen", a song by Bad4Good from the 1992 album ''Refugee'' * "Nineteen", a song by Karma to Burn from the 2001 album ''Almost Heathen''. * "Nineteen" (song), a 2007 song by American singer Billy Ray Cyrus. * "Nineteen", a song by Tegan and Sara from the 2007 album '' The Con''. * "XIX" (song), a 2014 song by Slipkno ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1841 Births
Events January–March * January 20 – Charles Elliot of the United Kingdom, and Qishan of the Qing dynasty, agree to the Convention of Chuenpi. * January 26 – Britain occupies Hong Kong. Later in the year, the first census of the island records a population of about 7,500. * January 27 – The active volcano Mount Erebus in Antarctica is discovered, and named by James Clark Ross. * January 28 – Ross discovers the "Victoria Barrier", later known as the Ross Ice Shelf. On the same voyage, he discovers the Ross Sea, Victoria Land and Mount Terror. * January 30 – A fire ruins and destroys two-thirds of the villa (modern-day city) of Mayagüez, Puerto Rico. * February 4 – First known reference to Groundhog Day in North America, in the diary of a James Morris. * February 10 – The Act of Union (''British North America Act'', 1840) is proclaimed in Canada. * February 11 – The two colonies of the Canadas are merged, into the United Province of Canada. * February ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kiel
Kiel () is the capital and most populous city in the northern Germany, German state of Schleswig-Holstein, with a population of 246,243 (2021). Kiel lies approximately north of Hamburg. Due to its geographic location in the southeast of the Jutland peninsula on the southwestern shore of the Baltic Sea, Kiel has become one of Germany's major maritime centres, known for a variety of international sailing events, including the annual Kiel Week, which is the biggest sailing event in the world. Kiel is also known for the Kiel mutiny, Kiel Mutiny, when sailors refused to board their vessels in protest against Germany's further participation in World War I, resulting in the abdication of the Wilhelm II, German Emperor, Kaiser and the formation of the Weimar Republic. The Olympic sailing competitions of the 1936 Summer Olympics, 1936 and the 1972 Summer Olympics#Venues, 1972 Summer Olympics were held in the Bay of Kiel. Kiel has also been one of the traditional homes of the German Nav ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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