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Hadjicostas's Formula
In mathematics, Hadjicostas's formula is a formula relating a certain double integral to values of the gamma function and the Riemann zeta function. It is named after Petros Hadjicostas. Statement Let ''s'' be a complex number with ''s'' ≠ -1 and Re(''s'') > −2. Then :\int_0^1\int_0^1 \frac(-\log(xy))^s\,dx\,dy=\Gamma(s+2)\left(\zeta(s+2)-\frac\right). Here Γ is the Gamma function and ζ is the Riemann zeta function. Background The first instance of the formula was proved and used by Frits Beukers in his 1978 paper giving an alternative proof of Apéry's theorem. He proved the formula when ''s'' = 0, and proved an equivalent formulation for the case ''s'' = 1. This led Petros Hadjicostas to conjecture the above formula in 2004, and within a week it had been proven by Robin Chapman. He proved the formula holds when Re(''s'') > −1, and then extended the result by analytic continuation to get the full result ... [...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 poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
