Appell Series

	Appell Series In mathematics, Appell series are a set of four hypergeometric series ''F''1, ''F''2, ''F''3, ''F''4 of two variables that were introduced by and that generalize Gauss's hypergeometric series 2''F''1 of one variable. Appell established the set of partial differential equations of which these functions are solutions, and found various reduction formulas and expressions of these series in terms of hypergeometric series of one variable. Definitions The Appell series ''F''1 is defined for , ''x'', < 1, , ''y'', < 1 by the double series : $F_1(a,b_1,b_2;c;x,y) = \sum_^\infty \frac \,x^m y^n ~,$ where $(q)_n$ is the Pochhammer symbol . For other values of ''x'' and ''y'' the function ''F''₁ can be defined by [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Appell–Lerch Series In mathematics, a mock modular form is the Holomorphic function, holomorphic part of a harmonic weak Maass wave form, Maass form, and a mock theta function is essentially a mock modular form of weight . The first examples of mock theta functions were described by Srinivasa Ramanujan in his last 1920 letter to G. H. Hardy and in his Ramanujan's lost notebook, lost notebook. Sander Zwegers discovered that adding certain non-holomorphic functions to them turns them into harmonic weak Maass forms. History Ramanujan's 12 January 1920 letter to Hardy listed 17 examples of functions that he called mock theta functions, and his lost notebook contained several more examples. (Ramanujan used the term "theta function" for what today would be called a modular form.) Ramanujan pointed out that they have an asymptotic expansion at the cusps, similar to that of modular forms of weight , possibly with poles at cusps, but cannot be expressed in terms of "ordinary" theta functions. He called fun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Integral In 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 ..., an integral assigns numbers to functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with Derivative, differentiation, integration is a fundamental, essential operation of calculus,Integral calculus is a very well established mathematical discipline for which there are many sources. See and , for example. and serves as a tool to solve problems in mathematics and physics involving the area of an arbitrary shape, the length of a curve, and the volume of a solid, among others. The integrals enumerated here are those termed definite integrals, which can be int ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]