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Rankin–Cohen Bracket
In mathematics, the Rankin–Cohen bracket of two modular form In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the Group action (mathematics), group action of the modular group, and also satisfying a grow ...s is another modular form, generalizing the product of two modular forms. gave some general conditions for polynomials in derivatives of modular forms to be modular forms, and found the explicit examples of such polynomials that give Rankin–Cohen brackets. They were named by , who introduced Rankin–Cohen algebras as an abstract setting for Rankin–Cohen brackets. Definition If f(\tau) and g(\tau) are modular forms of weight ''k'' and ''h'' respectively then their ''n''th Rankin–Cohen bracket [''f'',''g'']''n'' is given by :[f,g]_n = \frac\sum_ (-1)^r\binom\binom \frac\frac \ . It is a modular form of weight ''k'' + ''h'' + 2''n''. Note t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Form
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the Group action (mathematics), group action of the modular group, and also satisfying a growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections with number theory. Modular forms appear in other areas, such as algebraic topology, sphere packing, and string theory. A modular function is a function that is invariant with respect to the modular group, but without the condition that be Holomorphic function, holomorphic in the upper half-plane (among other requirements). Instead, modular functions are Meromorphic function, meromorphic (that is, they are holomorphic on the complement of a set of isolated points, which are poles of the function). Modular form theory is a special case of the more general theory of automorphic form ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |