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Faxén Integral
In mathematics, the Faxén integral (also named Faxén function) is the following integral :\operatorname(\alpha,\beta;x)=\int_0^ \exp(-t+xt^)t^\mathrmt,\qquad (0\leq \operatorname(\alpha) 0). The integral is named after the Swedish physicist Olov Hilding Faxén, who published it in 1921 in his PhD thesis. ''n''-dimensional Faxén integral More generally one defines the ''n-dimensional Faxén integral'' as :I_n(x)=\lambda_n\int_0^\cdots \int_0^t_1^\cdots t_n^e^\mathrmt_1\cdots \mathrmt_n, with :f(t_1,\dots,t_n;x):=\sum\limits_^n t_j^-xt_1^\cdots t_n^\quad and \quad\lambda_n:=\prod\limits_^n\mu_j for x \in \C and :(0<\alpha_i <\mu_i,\;\operatorname(\beta_i)>0,\; i=1,\dots,n). The parameter is only for convenience in calculations. Properties Let denote theGamma function
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Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral
In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus,Integral calculus is a very well established mathematical discipline for which there are many sources. See and , for example. the other being Derivative, differentiation. Integration was initially used to solve problems in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity. Usage of integration expanded to a wide variety of scientific fields thereafter. A definite integral computes the signed area of the region in the plane that is bounded by the Graph of a function, graph of a given Function (mathematics), function between two points in the real line. Conventionally, areas above the horizontal Coordinate axis, axis of the plane are positive while areas below are n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Olov Hilding Faxén
Olov (or Olof) is a Swedish form of Olav/Olaf, meaning "ancestor's descendant". A common short form of the name is ''Olle''. The name may refer to: * Olle Åberg (1925–2013), Swedish middle-distance runner * Olle Åhlund (1920–1996), Swedish footballer * Olle Anderberg (1919–2003), Swedish wrestler in the Olympic Games * Olle Andersson (speedway rider) (1932–2017), Swedish speedway rider * Olle Andersson (tennis) (1895–1974), Swedish tennis player * Olov Englund (born 1983), Swedish bandy player * Olof Forssberg (1938–2023), Swedish jurist and civil servant * Olle Hagnell (1924–2011), Swedish psychiatrist * Olle Hellbom (1925–1982), Swedish film director * Olof Johansson (born 1937), Swedish politician * Olov Lambatunga, Archbishop of Uppsala, Sweden, 1198–1206 * Olle Larsson (1928–1960), Swedish rower * Olof Mellberg (born 1977), Swedish footballer * Olof Mörck (born 1981), Swedish guitarist and songwriter, member of Amaranthe * Olle Nordemar (1914� ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uppsala University
Uppsala University (UU) () is a public university, public research university in Uppsala, Sweden. Founded in 1477, it is the List of universities in Sweden, oldest university in Sweden and the Nordic countries still in operation. Initially founded in the 15th century, the university rose to significance during the rise of Swedish Empire, Sweden as a great power at the end of the 16th century and was then given relative financial stability with a large donation from Monarchy of Sweden, King Gustavus Adolphus of Sweden, Gustavus Adolphus in the early 17th century. Uppsala also has an important historical place in Swedish national culture, and national identity, identity for the Swedish establishment: in historiography, religion, literature, politics, and music. Many aspects of Swedish academic culture in general, such as the white student cap, originated in Uppsala. It shares some peculiarities, such as the student nation system, with Lund University and the University of Helsink ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gamma Function
In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined for all complex numbers z except non-positive integers, and for every positive integer z=n, \Gamma(n) = (n-1)!\,.The gamma function can be defined via a convergent improper integral for complex numbers with positive real part: \Gamma(z) = \int_0^\infty t^ e^\textt, \ \qquad \Re(z) > 0\,.The gamma function then is defined in the complex plane as the analytic continuation of this integral function: it is a meromorphic function which is holomorphic function, holomorphic except at zero and the negative integers, where it has simple Zeros and poles, poles. The gamma function has no zeros, so the reciprocal gamma function is an entire function. In fact, the gamma function corresponds to the Mellin transform of the negative exponential functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scorer Function
In mathematics, the Scorer's functions are special functions studied by and denoted Gi(''x'') and Hi(''x''). Hi(''x'') and -Gi(''x'') solve the equation :y''(x) - x\ y(x) = \frac and are given by :\mathrm(x) = \frac \int_0^\infty \sin\left(\frac + xt\right)\, dt, :\mathrm(x) = \frac \int_0^\infty \exp\left(-\frac + xt\right)\, dt. The Scorer's functions can also be defined in terms of Airy function In the physical sciences, the Airy function (or Airy function of the first kind) is a special function named after the British astronomer George Biddell Airy (1801–1892). The function Ai(''x'') and the related function Bi(''x''), are Linear in ...s: :\begin \mathrm(x) &= \mathrm(x) \int_x^\infty \mathrm(t) \, dt + \mathrm(x) \int_0^x \mathrm(t) \, dt, \\ \mathrm(x) &= \mathrm(x) \int_^x \mathrm(t) \, dt - \mathrm(x) \int_^x \mathrm(t) \, dt. \end It can also be seen, just from the integral forms, that the following relationship holds: :\mathrm(x)+\mathrm(x)\equiv \math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (mathematics), series, and analytic functions. These theories are usually studied in the context of Real number, real and Complex number, complex numbers and Function (mathematics), functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any Space (mathematics), space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space). History Ancient Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functions And Mappings
In mathematics, a map or mapping is a function in its general sense. These terms may have originated as from the process of making a geographical map: ''mapping'' the Earth surface to a sheet of paper. The term ''map'' may be used to distinguish some special types of functions, such as homomorphisms. For example, a linear map is a homomorphism of vector spaces, while the term linear function may have this meaning or it may mean a linear polynomial. In category theory, a map may refer to a morphism. The term ''transformation'' can be used interchangeably, but '' transformation'' often refers to a function from a set to itself. There are also a few less common uses in logic and graph theory. Maps as functions In many branches of mathematics, the term ''map'' is used to mean a function, sometimes with a specific property of particular importance to that branch. For instance, a "map" is a "continuous function" in topology, a "linear transformation" in linear algebra, etc. So ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
