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Kummer's Function
In mathematics, there are several functions known as Kummer's function. One is known as the confluent hypergeometric function of Kummer. Another one, defined below, is related to the polylogarithm. Both are named for Ernst Kummer Ernst Eduard Kummer (29 January 1810 – 14 May 1893) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned .... Kummer's function is defined by :\Lambda_n(z)=\int_0^z \frac\;dt. The duplication formula is :\Lambda_n(z)+\Lambda_n(-z)= 2^\Lambda_n(-z^2). Compare this to the duplication formula for the polylogarithm: :\operatorname_n(z)+\operatorname_n(-z)= 2^\operatorname_n(z^2). An explicit link to the polylogarithm is given by :\operatorname_n(z)=\operatorname_n(1)\;\;+\;\; \sum_^ (-)^ \;\frac \;\operatorname_ (z) \;\;+\;\; \frac \;\left \Lambda_n(-1) - \Lambda_n(-z) \right References *. Special functio ... [...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 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Confluent Hypergeometric Function
In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular singularity. The term ''confluent'' refers to the merging of singular points of families of differential equations; ''confluere'' is Latin for "to flow together". There are several common standard forms of confluent hypergeometric functions: * Kummer's (confluent hypergeometric) function , introduced by , is a solution to Kummer's differential equation. This is also known as the confluent hypergeometric function of the first kind. There is a different and unrelated Kummer's function bearing the same name. * Tricomi's (confluent hypergeometric) function introduced by , sometimes denoted by , is another solution to Kummer's equation. This is also known as the confluent hypergeometric function of the second kind. * Whittaker functions (fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polylogarithm
In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function of order and argument . Only for special values of does the polylogarithm reduce to an elementary function such as the natural logarithm or a rational function. In quantum statistics, the polylogarithm function appears as the closed form of integrals of the Fermi–Dirac distribution and the Bose–Einstein distribution, and is also known as the Fermi–Dirac integral or the Bose–Einstein integral. In quantum electrodynamics, polylogarithms of positive integer order arise in the calculation of processes represented by higher-order Feynman diagrams. The polylogarithm function is equivalent to the Hurwitz zeta function — either function can be expressed in terms of the other — and both functions are special cases of the Lerch transcendent. Polylogarithms should not be confused with polylogarithmic functions nor with the offset logarithmic integral ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ernst Kummer
Ernst Eduard Kummer (29 January 1810 – 14 May 1893) was a German 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 .... Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a ''Gymnasium (school), gymnasium'', the German equivalent of high school, where he inspired the mathematical career of Leopold Kronecker. Life Kummer was born in Sorau, Province of Brandenburg, Brandenburg (then part of Prussia). He was awarded a PhD from the University of Halle in 1831 for writing a prize-winning mathematical essay (''De cosinuum et sinuum potestatibus secundum cosinus et sinus arcuum multiplicium evolvendis''), which was eventually published a year later. In 1840, Kummer married Ottilie Mendelssohn, daughter of N ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Duplication Formula
Duplication, duplicate, and duplicator may refer to: Biology and genetics * Gene duplication, a process which can result in free mutation * Chromosomal duplication, which can cause Bloom and Rett syndrome * Polyploidy, a phenomenon also known as ''ancient genome duplication'' * Enteric duplication cysts, certain portions of the gastrointestinal tract * Diprosopus, a form of cojoined twins also known as ''craniofacial duplication'' * Diphallia, a medical condition also known as ''penile duplication'' Computing * Duplicate code, a source code sequence that occurs more than once in a program * Duplicate characters in Unicode, pairs of single Unicode code points that are canonically equivalent. The reason for this are compatibility issues with legacy systems * Data redundancy, either wanted or unwanted (in which case one resorts to data deduplication) * Content copying through cut, copy, and paste * File copying Mathematics * Duplication matrix, a linear transformation d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Special Functions
Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defined by consensus, and thus lacks a general formal definition, but the List of mathematical functions contains functions that are commonly accepted as special. Tables of special functions Many special functions appear as solutions of differential equations or integrals of elementary functions. Therefore, tables of integrals usually include descriptions of special functions, and tables of special functions include most important integrals; at least, the integral representation of special functions. Because symmetries of differential equations are essential to both physics and mathematics, the theory of special functions is closely related to the theory of Lie groups and Lie algebras, as well as certain topics in mathematical physics. Symbol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
