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Chowla–Selberg Formula
In mathematics, the Chowla–Selberg formula is the evaluation of a certain product of values of the gamma function at rational values in terms of values of the Dedekind eta function at imaginary quadratic irrational numbers. The result was essentially found by and rediscovered by . Statement In logarithmic form, the Chowla–Selberg formula states that in certain cases the sum : \frac\sum_r \chi(r)\log \Gamma\left( \frac \right) = \frac\log(4\pi\sqrt) +\sum_\tau\log\left(\sqrt, \eta(\tau), ^2\right) can be evaluated using the Kronecker limit formula. Here χ is the quadratic residue symbol modulo ''D'', where ''−D'' is the discriminant of an imaginary quadratic field. The sum is taken over 0 < ''r'' < ''D'', with the usual convention χ(''r'') = 0 if ''r'' and ''D'' have a common factor. The function η is the Dedekind eta function, and ''h'' is the class number, and ''w'' is the number of roots of unity. Origin and applications The origin of such formu ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
