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Incomplete Fermi–Dirac Integral
In mathematics, the incomplete Fermi–Dirac integral for an index ''j'' is given by :F_j(x,b) = \frac \int_b^\infty \frac\,dt. This is an alternate definition of the incomplete polylogarithm. See also * Complete Fermi–Dirac integral In mathematics, the complete Fermi–Dirac integral, named after Enrico Fermi and Paul Dirac, for an index ''j '' is defined by :F_j(x) = \frac \int_0^\infty \frac\,dt, \qquad (j > -1) This equals :-\operatorname_(-e^x), where \operatornam ... External links GNU Scientific Library - Reference Manual Special functions {{mathanalysis-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Enrico Fermi
Enrico Fermi (; 29 September 1901 – 28 November 1954) was an Italian (later naturalized American) physicist and the creator of the world's first nuclear reactor, the Chicago Pile-1. He has been called the "architect of the nuclear age" and the "architect of the atomic bomb". He was one of very few physicists to excel in both theoretical physics and experimental physics. Fermi was awarded the 1938 Nobel Prize in Physics for his work on induced radioactivity by neutron bombardment and for the discovery of transuranium elements. With his colleagues, Fermi filed several patents related to the use of nuclear power, all of which were taken over by the US government. He made significant contributions to the development of statistical mechanics, Quantum mechanics, quantum theory, and nuclear physics, nuclear and particle physics. Fermi's first major contribution involved the field of statistical mechanics. After Wolfgang Pauli formulated his Pauli exclusion principle, exclusion pri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Dirac
Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the University of Cambridge, a professor of physics at Florida State University and the University of Miami, and a 1933 Nobel Prize recipient. Dirac made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics. Among other discoveries, he formulated the Dirac equation which describes the behaviour of fermions and predicted the existence of antimatter. Dirac shared the 1933 Nobel Prize in Physics with Erwin Schrödinger "for the discovery of new productive forms of atomic theory". He also made significant contributions to the reconciliation of general relativity with quantum mechanics. Dirac was regarded by his friends and colleagues as unusual in character. In a 1926 letter to Paul Ehrenfest, Albert ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Incomplete Polylogarithm
In mathematics, the Incomplete Polylogarithm function is related to the polylogarithm function. It is sometimes known as the incomplete Fermi–Dirac integral or the incomplete Bose–Einstein integral. It may be defined by: : \operatorname_s(b,z) = \frac\int_b^\infty \frac~dx. Expanding about z=0 and integrating gives a series representation: : \operatorname_s(b,z) = \sum_^\infty \frac~\frac where Γ(s) is the gamma function and Γ(s,x) is the upper incomplete gamma function In mathematics, the upper and lower incomplete gamma functions are types of special functions which arise as solutions to various mathematical problems such as certain integrals. Their respective names stem from their integral definitions, which .... Since Γ(s,0)=Γ(s), it follows that: : \operatorname_s(0,z) =\operatorname{Li}_s(z) where Lis(.) is the polylogarithm function. References * GNU Scientific Library - Reference Manual https://www.gnu.org/software/gsl/manual/gsl-re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Fermi–Dirac Integral
In mathematics, the complete Fermi–Dirac integral, named after Enrico Fermi and Paul Dirac, for an index ''j '' is defined by :F_j(x) = \frac \int_0^\infty \frac\,dt, \qquad (j > -1) This equals :-\operatorname_(-e^x), where \operatorname_(z) is the polylogarithm. Its derivative is :\frac = F_(x) , and this derivative relationship is used to define the Fermi-Dirac integral for nonpositive indices ''j''. Differing notation for F_j appears in the literature, for instance some authors omit the factor 1/\Gamma(j+1). The definition used here matches that in thNIST DLMF Special values The closed form of the function exists for ''j'' = 0: :F_0(x) = \ln(1+\exp(x)). For ''x = 0'', the result reduces to F_j(0) = \eta(j+1), where \eta is the Dirichlet eta function. See also * Incomplete Fermi–Dirac integral * Gamma function * Polylogarithm In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
